On the non-collapsed RCD spaces with local bounded covering geometry

Jikang Wang UC Berkeley, Berkeley, CA, US [email protected]

Abstract.

We consider a RCD $(-(N-1),N)$ space $(X,d,\mathcal{H}^{N})$ with local bounded covering geometry. The first result is related to Gromov’s almost flat manifold theorem. Specifically, if for every point $\tilde{p}$ in the universal cover $\widetilde{X}$ , we have $\mathcal{H}^{N}(B_{1}(\tilde{p}))\geq v>0$ and the diameter of $X$ is sufficiently small, then $X$ is biHölder homeomorphic to an infranil-manifold. Moreover, if $X$ is a smooth Riemannian $N$ -manifold with $\mathrm{Ric}\geq-(N-1)$ , then $X$ is biHölder diffeomorphic to an infranil-manifold. An application of our argument is to confirm the conjecture that Gromov’s almost flat manifold theorem holds in the $\mathrm{RCD}+\mathrm{CBA}$ setting.

The second result concerns a regular fibration theorem. Let $(X_{i},d_{i},\mathcal{H}^{N})$ be a sequence of RCD $(-(N-1),N)$ spaces converging to a compact smooth $k$ -dimensional manifold $K$ in the Gromov-Hausdorff sense. Assume that for any $p_{i}\in X_{i}$ , the local universal cover is non-collapsing, i.e., for any pre-image point $\tilde{p}_{i}$ of $p_{i}$ in the universal cover of the ball $B_{3}(p_{i})$ , we have $\mathcal{H}^{N}(B_{1}(\tilde{p}_{i}))\geq v$ for some fixed $v>0$ . Then for sufficiently large $i$ , there exists a fibration map $f_{i}:X_{i}\to K$ , where the fiber is an infra-nilmanifold and the structure group is affine.

Dedicated to Xiaochun Rong’s 70th Birthday

1. Introduction

In this paper, we study the topology of a RCD $(-(N-1),N)$ space $(X,d,\mathcal{H}^{N})$ which satisfies $(\rho,v)$ -bound covering condition, i.e., for any $p\in X$ , take the universal cover of $\rho$ -ball $B_{\rho}(p)$ and $\tilde{p}$ is a pre-image of $p$ , then $\mathcal{H}^{N}(B_{\rho/3}(\tilde{p}))\geq v$ , where $\rho,v>0$ .

We first review the topological theory of collapsing manifolds with sectional curvature bound. The one of most important theorems is Gromov’s almost flat theorem.

Theorem 1.1 (Gromov’s almost flat manifold theorem, [20, 34, 36]).

Given $n\in\mathbb{N}$ , there exists $\epsilon(n)>0$ and $C(n)>0$ so that for any compact almost flat $n$ -manifold $M$ with

\max(|\sec_{M}|)\mathrm{diam}(M)^{2}<\epsilon(n),

Then $M$ is diffeomorphic to an infra-nilmanifold, $\mathcal{N}/\Gamma$ , where $\mathcal{N}$ is a simply connected nilpotent $n$ -dim Lie group and $\Gamma$ is a discrete subgroup of $\mathcal{N}\rtimes\text{Aut}(\mathcal{N})$ and $[\Gamma:\Gamma\cap\mathcal{N}]\leq C(n)$ .

A related result, due to Fukaya, asserts that if a sequence of $n$ -manifolds $M_{i}$ with $|\sec|\leq 1$ converging to a lower dimensional manifold $K$ , then there is a fibration map $f_{i}:M_{i}\to K$ where the fibers are infra-nilmanifolds with an affine structure group.

Theorem 1.2 (smooth fibration, [16, 9]).

Assume that a sequence of $n$ -manifolds $M_{i}$ with $|\sec_{M_{i}}|\leq 1$ converges to a compact lower dimensional manifold $K$ in the Gromov-Hausdorff sense, $M_{i}\overset{GH}{\longrightarrow}K$ . Then for $i$ large enough, there is a smooth fiberation map $f_{i}:M_{i}\to N$ with fiber an infra-nilmanifold and an affine structure group, and $f_{i}$ is a Gromov-Hausdorff approximation (GHA).

More generally, the theories of singular fibration and nilpotent killing structure for collapsed manifolds with $|\sec|\leq 1$ have been extensively studied in [9, 10, 11, 16, 17, 18]. However, Theorems 1.1 and 1.2 may not hold if we replace the sectional curvature by the Ricci curvature [2].

Around 2016, Rong proposed to investigate the class of $n$ -dim manifolds $M$ satisfying $(\rho,v)$ -bound Ricci covering geometry. Specifically, $\mathrm{Ric}_{M}\geq-(n-1)$ , and for any point $p\in M$ , take the universal cover of the $\rho$ -ball $B_{\rho}(p)$ with $\tilde{p}$ a pre-image of $p$ . Then the $\mathrm{Vol}(B_{\rho/3}(\tilde{p}))\geq v$ . According to [9], any $n$ -manifold with $|\mathrm{sec}|\leq 1$ satisfies the $(\rho,v)$ -bound Ricci covering condition for some $\rho,v>0$ depending on $n$ .

Theorems 1.1 and 1.2 can be generalized to $(\rho,v)$ -bound Ricci covering geometry, see Theorem 1.3 and 1.8. If the diameter of $M$ is less than $\rho$ , then $(\rho,v)$ -bound covering condition is exactly that the universal cover $\widetilde{M}$ is non-collapsing.

Theorem 1.3 ([24, 35]).

Given $n,v>0$ , there exists $\epsilon(n,v)>0$ and $C(n)>0$ , so that if a $n$ -manifold $M$ satisfies:

\mathrm{Ric}\geq-(n-1),\ \mathrm{diam}(M)<\epsilon\leq\epsilon(n,v),\ \mathrm{Vol}(B_{1}(\tilde{p}))\geq v,\ \forall\tilde{p}\in\widetilde{M},

then $M$ is diffeomorphic to an infra-nilmanifold.

We summarize the proofs of Theorem 1.3 as follows. In [24], it was proved that the Ricci flow on $M$ exists for a definite time. After running the flow, we get an almost flat metric on $M$ . Then we can apply Gromov’s almost flat manifold theorem. The proof in [35] uses successively blowing up technique and the structure of iterated bundles, avoiding reliance on Gromov’s result.

Recently, Zamora and Zhu proved the topology rigidity for a RCD $(K,N)$ space with a small diameter, extending previous work in [28]. By the generalized Margulis lemma (Theorem 2.4), if the diameter of a RCD $(K,N)$ space $(X,p)$ is sufficiently small, then $\pi_{1}(X,p)$ contains a nilpotent subgroup $G$ with index $\leq C(N)$ . Then we can find a descending sequence

G=G_{1}\vartriangleright G_{2}\vartriangleright...\vartriangleright G_{k}=\{e\}

such that $G_{i}/G_{i+1}$ is cyclic, $i=1,2...,k-1$ . Then we define the rank of $\pi_{1}(X)$ is the number of $i$ such that $G_{i}/G_{i+1}$ is infinite.

Theorem 1.4.

([45]) For any $K\in\mathbb{R}$ and $N\geq 1$ , there exists $\epsilon>0$ such that for any RCD $(K,N)$ space $(X,d,\mathfrak{m})$ with diameter less than $\epsilon$ , then rank $(\pi_{1}(X))\leq N$ . Moreover, if rank $(\pi_{1}(X))=N$ , then $X$ is homeomorphic to an infranil-manifold of dimension $N$ .

Zamora and Zhu conjectured that the homeomorphism in Theorem 1.4 can be biHölder.

We call a RCD $(-(N-1),N)$ space non-collapsed if the measure is the Hausdorff measure $\mathcal{H}^{N}$ ; in particular, $N\in\mathbb{N}$ . We say that a sequence of RCD $(-(N-1),N)$ spaces $(X_{i},d_{i},\mathcal{H}^{N})$ are non-collapsing if there exists $v>0$ such that for any $p_{i}\in(X_{i},d_{i},\mathcal{H}^{N})$ , $\mathcal{H}^{N}(B_{1}(p_{i}))\geq v$ . Sometimes, we simply say a RCD $(-(N-1),N)$ space $(X,d,\mathcal{H}^{N})$ non-collapsing if the Hausdorff measure of any one ball in $X$ is bounded below by a fixed number.

Remark 1.5.

Consider a RCD $(-(N-1),N)$ space $(X,d,\mathcal{H}^{N})$ with a small diameter. By [45], if rank $(\pi_{1}(X))=N$ then the universal cover $\widetilde{X}$ is non-collapsing. Conversely, if the universal cover $\widetilde{X}$ is non-collapsing, then rank $(\pi_{1}(X))=N$ by our proof of Theorem A.

At present, the proofs of Theorem 1.3 and the rigidity part of Theorem 1.4 are different. The proofs in [24, 35] for Theorem 1.3 rely on the smooth structure, and hence cannot be directly extended to the non-smooth setting. On the other hand, the proof in [45] for the rigidity part of Theorem 1.4 uses a topological result for aspherical manifolds to find the homeomorphism, but does not prove diffeomorphism in the smooth case. The first main result in this paper is to give a new proof that works for both Theorem 1.3 and the rigidity part of Theorem 1.4. Further, we can show that the homeomorphism is biHölder, thereby confirming the conjecture in [45].

Theorem A.

Given $N,v>0$ , there exists $\epsilon(N,v)>0$ and $C(N)>0$ , so that if a RCD $(-(N-1),N)$ space $(X,d,\mathcal{H}^{N})$ satisfies:

\mathrm{diam}(X)<\epsilon\leq\epsilon(N,v),\ \mathcal{H}^{N}(B_{1}(\tilde{p}))\geq v,\ \forall\tilde{p}\in\widetilde{X},

then $X$ is biHölder homeomorphic to an infranil-manifold $\mathcal{N}/\Gamma$ where $\mathcal{N}$ has a left-invariant metric. More precisely, there exists $f:X\to\mathcal{N}/\Gamma$ such that for all $x,y\in X$ ,

(1-\Phi(\epsilon|N,v))d(x,y)^{1+\Phi(\epsilon|N,v)}\leq d(f(x),f(y))\leq(1+\Phi(\epsilon|N,v))d(x,y),

where $\Phi(\epsilon|N,v)\to 0$ as $\epsilon\to 0$ . Moreover, if $X$ is a smooth $N$ -manifold with $\mathrm{Ric}\geq-(N-1)$ , then $f$ is a diffeomorphism.

The proof of Theorem A can be extended to metric spaces with mixed curvature. Kapovitch showed that Gromov’s almost flat manifolds theorem holds for weighted closed manifolds with upper sectional and lower Bakry–Emery Ricci curvature bounds.

Theorem 1.6 ([26]).

For any $1<N<\infty$ , there exists $\epsilon>0$ such that the following hold. If $(M,g,e^{-f}\mathcal{H}^{n})$ is a weighted closed Riemannian $n$ -manifold with $n\leq N$ , $\mathrm{sec}\leq\epsilon$ , $\mathrm{diam}\leq\epsilon$ and $\mathrm{Ric}_{f,N}\geq-\epsilon$ , then $M$ is diffeomorphic to an infranil-manifold.

We briefly recall the proof of Theorem 1.6. Kapovitch first proved that $M$ is aspherical by a fibration theorem and an induction argument. By some topological results, $M$ is homeomorphic to an infranil-manifold. Then applying Ricci flow smoothing techniques and Gromov’s almost flat theorem, $M$ is diffeomorphic to an infranil-manifold.

It was conjectured in [26] that Gromov’s almost flat manifold theorem also holds under the $\mathrm{RCD}+\mathrm{CBA}$ conditions. We recall the structure theory of $\mathrm{RCD}+\mathrm{CBA}$ spaces from [27]. If $(X,d,\mathfrak{m})$ is $\mathrm{RCD}(K,N)$ and $\mathrm{CBA}(k)$ with $N<\infty$ then $X$ is a topological manifold with boundary of dimension $\leq N$ . The manifold part of $X$ is a smooth $C^{1}$ -manifold with a $C^{0}\cap\mathrm{BV}$ Riemannian metric which induces the distance function $d$ on $X$ . In particular, if $\partial X=\emptyset$ , then $X$ is a smooth manifold.

Modify the proof Theorem A a little bit, we can prove the following conjecture in [26].

Theorem 1.7.

For any $1<N<\infty$ there exists $\epsilon(N)$ such that for any $\epsilon<\epsilon(N)$ the following holds. If $(X,d,\mathfrak{m})$ is an $\mathrm{RCD}(-\epsilon,N)$ space such that $(X,d)$ is $\mathrm{CBA}(\epsilon)$ , $\partial X=\emptyset$ and $\mathrm{diam}(X)\leq\epsilon$ , then $X$ is biLipschitz diffeomorphic to an infranil-manifold of dimension $\leq N$ . That is, there exist a infranil-manifold $\mathcal{N}/\Gamma$ and a diffeomorphic map $f:X\to\mathcal{N}/\Gamma$ so that

(1-\Phi(\epsilon|N))d(x,y)\leq d(f(x),f(y))\leq(1+\Phi(\epsilon|N))d(x,y),

where $\Phi(\epsilon|N)\to 0$ as $\epsilon\to 0$ .

It was known in [26] that Theorem 1.7 holds up to homeomorphism, while it remains unknown whether we can apply Ricci flow to smooth the metric. In particular, the proof for Theorem 1.7 provides an alternative approach to proving Theorem 1.6, without relying on Ricci flow smoothing techniques or Gromov’s almost flat manifold theorem.

Our next result is to prove a regular fibraion theorem in the RCD setting. The following fibration theorem in the smooth case was proved by Huang and Rong.

Theorem 1.8 ([23, 35]).

Given $n,v>0$ , there exists $\epsilon(n,v)>0$ and $C(n)>0$ so that for any compact $n$ -manifold $M$ and $k$ -manifold $K$ satisfying:

$\text{Ric}_{M}\geq-(n-1)$ , $(1,v)$ -bound covering geometry holds on $M$ ,

$|\text{sec}_{K}|\leq 1$ , $\text{inj}_{K}\geq 1$ , $d_{GH}(M,K)<\epsilon<\epsilon(n,v)$ ,

then there is a smooth fiber bundle map, $f:M\to K$ that is a $\Phi(\epsilon|n,v)$ -Gromov-Hausdorff approximation (GHA), where $\Phi(\epsilon|n,v)\to 0$ as $\epsilon\to 0$ . The fiber is an infranil-manifold and the structure group is affine

Huang first constructed the fibration in [23], and Rong proved that the fiber is an infranil-manifold and that the structure group be can be reduced to be affine in [35].

We generalize Theorem 1.8 to RCD $(-(N-1),N)$ spaces $(X,d,\mathcal{H}^{N})$ with $(\rho,v)$ -bound covering geometry.

Theorem B.

Given $N,v>0$ , suppose that a sequence of compact RCD $(-(N-1),N)$ spaces $(X_{i},d_{i},\mathcal{H}^{N})$ , with $(1,v)$ -bound covering geometry, converges to a compact smooth $k$ -manifold $K$ in the Gromov-Hausdorff sense. Then for sufficiently large $i$ , there exists a fiber bundle map $f_{i}:X_{i}\to K$ that is an $\epsilon_{i}$ -Gromov-Hausdorff approximation(GHA) where $\epsilon_{i}\to 0$ as $i\to\infty$ . Moreover, the fiber is homeomorphic to an infranil-manifold and the structure group is affine.

Next, we study the limit of RCD $(-(N-1),N)$ spaces $(X_{i},d_{i},\mathcal{H}^{N})$ with $(\rho,v)$ -bound covering geometry. By [25, 46], for any $k\geq 3$ , there exists a collapsing Ricci limit space with the rectifiable dimension equal to $k$ , which contains no manifold points. We shall prove that, manifold points in the limit of $(X_{i},d_{i},\mathcal{H}^{N})$ with $(\rho,v)$ -bound covering geometry have full measure.

Theorem 1.9.

Assume that $(X_{i},d_{i},\mathcal{H}^{N},p_{i})$ is a sequence of pointed RCD $(-(N-1),N)$ spaces with $(1,v)$ -bound covering geometry, and suppose that

(X_{i},d_{i},\mathcal{H}^{N},p_{i})\overset{\mathrm{pmGH}}{\longrightarrow}(X,d,\mathfrak{m},p)

with rectifiable dim $(X)=k<N$ . Then
(1) ( $k$ -regular points are manifold points) For any $q\in\mathcal{R}^{k}(X)$ , a neighborhood of $q$ is biHölder to an open set in $\mathbb{R}^{k}$ ;
(2) (fibration near $k$ -regular point) If $q\in\mathcal{R}^{k}(X)$ and $q_{i}\in X_{i}$ converging to $q$ , for sufficiently large $i$ , a neighborhood $U_{i}$ is of $q_{i}$ is biHölder to $V\times\mathcal{N}_{i}/\Gamma_{i}$ where $V$ is a neighborhood of $q$ and $\mathcal{N}_{i}/\Gamma_{i}$ is an infranil-manifold.

Remark 1.10.

It was shown by Rong that in the setting of Theorem 1.9, the Hausdorff dimension of $(X,d)$ is equal to $k$ and any tangent cone of $X$ is a metric cone.

Now we sketch our proof of Theorem A. We proceed by contradiction. Suppose that there exists a sequence of RCD $(-(N-1),N)$ spaces $(X_{i},d_{i},\mathcal{H}^{N},p_{i})$ with non-collapsing universal covers and $\mathrm{diam}(X_{i})\to 0$ , while none of $X_{i}$ is biHölder to an infranil-manifold. By the generalized Margulis lemma, there is a nilpotent subgroup $G_{i}$ of $G_{i}^{\prime}=\pi_{1}(X_{i},p_{i})$ with index $\leq C(n)$ . We may blow up the metric slowly at a regular point if necessary, and assume that the following diagram holds:

$\begin{CD}(\widetilde{X}_{i},\tilde{p}_{i},G_{i},G_{i}^{\prime})@>{eGH}>{}>(\mathbb{R}^{N},\tilde{p},G,G^{\prime})\\ @V{}V{\pi}V@V{}V{\pi}V\\ \widetilde{X}_{i}/G_{i}@>{GH}>{}>\mathrm{pt}.\end{CD}$

We can show that $G$ actions are free translation actions, which can be identified as $\mathbb{R}^{N}$ . Therefore by the structure theorem of approximate groups [3, 44], a neighborhood of the identity in $G_{i}$ forms a nilprogression. Roughly speaking, a nilprogression is a subset of a lattice in a simply connected nilpotent Lie group, and the nilprogression contains all generators and relations of the lattice.

Since the diameter of $\widetilde{X}_{i}/G_{i}$ converges to $0$ , $G_{i}$ is determined by the neighborhood of the identity. This small nrighborhood contains all generators and relations of $G_{i}$ . Thus $G_{i}$ must be isomorphic to a lattice (the groupfication of the nilprogression) in a simply connected nilpotent $N$ -dim Lie group $\mathcal{N}_{i}$ , where the Lie algebra structure of $\mathcal{N}_{i}$ converges to the one of $\mathbb{R}^{N}$ . Hence we can endow $\mathcal{N}_{i}$ with a left-invariant metric, which pointed converges to the flat metric on $\mathbb{R}^{N}$ in the $C^{4}$ -sense.

Next we identify $G_{i}^{\prime}$ as a subgroup in $\mathcal{N}_{i}\rtimes\text{Aut}(\mathcal{N}_{i})$ by the rigidity. $(\widetilde{X}_{i},\tilde{p}_{i},G_{i}^{\prime})$ is eGH-close to $(\mathcal{N}_{i},e,G_{i}^{\prime})$ on the $\frac{1}{\epsilon}$ -ball of the base point for some fixed small $\epsilon>0$ . Thus by an extension lemma, we can construct a global map $h:\widetilde{X}_{i}\to\mathcal{N}_{i},$ which is almost $G_{i}^{\prime}$ -equivariant and a GHA on any $\frac{1}{\epsilon}$ -ball in $\widetilde{X}_{i}$ . Specifically, for any $\tilde{x}\in\widetilde{X}_{i}$ , $h:B_{r}(\tilde{x})\to\mathcal{N}_{i}$ is a $\epsilon$ -GHA to its image; $d(h(g\tilde{x}),gh(\tilde{x}))\leq\epsilon$ for any $g\in G_{i}^{\prime}$ and $\tilde{x}\in\widetilde{X}_{i}$ .

Then we can find a normal subgroup $G_{i}^{\prime\prime}$ of $G_{i}^{\prime}$ with finite index, so that $G_{i}^{\prime\prime}\subset G_{i}$ and $G_{i}^{\prime\prime}\cap B_{\frac{1}{\epsilon}}(e)=\emptyset$ where $e\in\mathcal{N}_{i}$ . Thus $(\widetilde{X}_{i}/G_{i}^{\prime\prime},G_{i}^{\prime}/G_{i}^{\prime\prime})$ is eGH close to $(\mathcal{N}_{i}/G_{i}^{\prime\prime},G_{i}^{\prime}/G_{i}^{\prime\prime})$ on any $\frac{1}{\epsilon}$ -ball. Then we apply an averaging technique (see Theorem 3.5) to obtain a $G_{i}^{\prime}/G_{i}^{\prime\prime}$ -equivariant map $f_{G_{i}^{\prime}/G_{i}^{\prime\prime}}$ from $\widetilde{X}_{i}/G_{i}^{\prime\prime}$ to $\mathcal{N}_{i}/G_{i}^{\prime\prime}$ , which is locally almost $N$ -splitting. By the canonical Reifenberg method from [12] and [22], $f_{G_{i}^{\prime}/G_{i}^{\prime\prime}}$ must be a biHöder homeomorphism. Since $f_{G_{i}^{\prime}/G_{i}^{\prime\prime}}$ is $G_{i}^{\prime}/G_{i}^{\prime\prime}$ -equivariant, it follows that $X_{i}=\widetilde{X}_{i}/G_{i}^{\prime}$ is biHölder homeomorphic to the infranil-manifold $\mathcal{N}_{i}/G_{i}^{\prime}$ . Then we finish the proof of Theorem A.

The smooth fibration map $f$ in Theorem 1.8 is constructed by averaging and gluing some local almost $k$ -splitting maps. Thus $f$ is a smooth GHA. Then Huang compared $f$ to a linear average and showed that $df$ is non-degenerate using the local bounded covering geometry. Therefore $f$ is a fibration map by the implicit function theorem.

In the non-smooth setting in Theorem B, we can construct a GHA $f_{i}:X_{i}\to K$ which is a locally almost $k$ -splitting map. Although there is no implicit function theorem in the non-smooth setting, we can prove that $f_{i}$ is a fibration with infranil-manifold fiber by applying the proof in Theorem A to the collapsing directions of $X_{i}$ . The structure group can be affine because the nilpotent structure of the fiber is independent of the choice of the base point; see also [35].

Remark 1.11.

Pointed Gromov-Hausdorff approximations typically cannot provide information about the global topology of non-compact spaces, as they do not capture the geometry outside of a large ball. This limitation is why we consider a global map $h$ which is almost equivariant and acts as a GHA on any $\frac{1}{\epsilon}$ -ball.

Acknowledgments. The author would like to thank Xiaochun Rong, Vitali Kapovitch, Jiayin Pan, Shicheng Xu, Xingyu Zhu and Sergio Zamora for helpful discussions.

2. Preliminaries

2.1. Equivariant Gromov-Hausdorff convergence and isometry group on a Ricci limit space

We review the notion of equivariant Gromov-Hausdorff convergence introduced by Fukaya and Yamaguchi [15, 19].

Let $(X,p)$ and $(Y,q)$ be two pointed metric spaces. Let $H$ and $K$ be closed subgroups of $\mathrm{Isom}(X)$ and $\mathrm{Isom}(Y)$ , respectively. For any $r>0$ , define the sets

H(p,r)=\{h\in H|d(hp,p)\leq r\},\ K(q,r)=\{k\in K|d(kq,q)\leq r\}.

For $\epsilon>0$ , a pointed $\epsilon$ -equivariant Gromov-Hausdorff approximation (or simply an $\epsilon$ -eGHA) is a triple of maps $(f,\phi,\psi)$ where:

f:B_{\frac{1}{\epsilon}}(p)\to B_{\frac{1}{\epsilon}+\epsilon}(q),\quad\phi:H(p,\frac{1}{\epsilon})\to K(q,\frac{1}{\epsilon}),\quad\psi:K(q,\frac{1}{\epsilon})\to H(p,\frac{1}{\epsilon})

satisfying the following conditions:
(1) $f(p)=q$ , $f(B_{\frac{1}{\epsilon}}(p))$ is $2\epsilon$ -dense in $B_{\frac{1}{\epsilon}+\epsilon}(q)$ and $|d(f(x_{1}),f(x_{2}))-d(x_{1},x_{2})|\leq\epsilon$ for all $x_{1},x_{2}\in B_{\frac{1}{\epsilon}}(p)$ ;
(2) $d(\phi(h)f(x),f(hx))<\epsilon$ for all $h\in H(\frac{1}{\epsilon})$ and $x\in B_{\frac{1}{\epsilon}}(p)$ ;
(3) $d(kf(x),f(\psi(k)x))<\epsilon$ for all $k\in K(\frac{1}{\epsilon})$ and $x\in B_{\frac{1}{\epsilon}}(p)$ .

The equivariant Gromov-Hausdorff(eGH) distance $d_{eGH}((X_{i},p_{i},G_{i}),(X,p,G))$ is defined as the infimum of $\epsilon$ so that there exists a $\epsilon$ -eGHA. A sequence of metric space with isometric actions $(X_{i},p_{i},G_{i})$ converges to a limit space $(X,p,G)$ , if $d_{eGH}((X_{i},p_{i},G_{i}),(X,p,G))\to 0$ .

Given a Gromov-Hausdorff approximation(GHA) $f$ as in condition (1) above, we can construct an admissible metric on the disjoint union $B_{\frac{1}{\epsilon}}(p)\sqcup B_{\frac{1}{\epsilon}}(q)$ so that

B_{\frac{1}{\epsilon}}(p)\hookrightarrow B_{\frac{1}{\epsilon}}(p)\sqcup B_{\frac{1}{\epsilon}}(q),B_{\frac{1}{\epsilon}}(q)\hookrightarrow B_{\frac{1}{\epsilon}}(p)\sqcup B_{\frac{1}{\epsilon}}(q)

are isometric embedding and for any $x\in B_{\frac{1}{\epsilon}}(p)$ , $d(x,f(x))\leq 2\epsilon$ . We always assume such an admissible metric whenever Gromov-Hausdorff distance between two metric spaces are small.

We have the following pre-compactness theorem for equivariant Gromov-Hausdorff convergence [15, 19].

Theorem 2.1 ([15, 19]).

Let $(X_{i},p_{i})$ be a sequence of metric spaces converging to a limit space $(X,p)$ in the pointed Gromov-Hausdorff sense. For each $i$ , let $G_{i}$ be a closed subgroup of $\mathrm{Isom}(X_{i})$ , the isometry group of $X_{i}$ . Then passing to a subsequence if necessary,

(X_{i},p_{i},G_{i})\overset{eGH}{\longrightarrow}(X,p,G),

where $G$ is a closed subgroup of $\mathrm{Isom}(X)$ . Moreover, the quotient spaces $(X_{i}/G_{i},\bar{p}_{i})$ pointed Gromov-Hausdorff converge to $(X/G,p)$ .

2.2. Geometric Theory of RCD $(K,N)$ spaces

In this subsection, we review the structure theory of RCD $(K,N)$ spaces. We assume that the reader is familiar with the basic notions of RCD spaces. A measured metric RCD $(K,N)$ space $(X,d,\mathfrak{m})$ is non-collapsed if $N\in\mathbb{N}$ and $\mathfrak{m}=\mathcal{H}^{N}$ .

We begin by defining regular points on an RCD space. We use the notation $0^{k}$ to refer the origin in $\mathbb{R}^{k}$ .

Definition 2.2.

Let $(X,d,\mathfrak{m})$ be an RCD $(K,N)$ space. Given $\epsilon>0$ , $r>0$ , and $k\in\mathbb{N}$ , we define

\mathcal{R}_{\epsilon,r}^{k}(X)=\{x\in X:d_{GH}(B_{s}(x),B_{s}(0^{k}))<\epsilon s,\forall\ 0<s<2r\},

where $0^{k}\in\mathbb{R}^{k}$ , and

\mathcal{R}_{\epsilon}^{k}(X)=\bigcup_{r>0}\mathcal{R}_{\epsilon,r}^{k}(X),\ \mathcal{R}^{k}(X)=\bigcap_{\epsilon>0}\mathcal{R}_{\epsilon}^{k}(X).

For any RCD $(K,N)$ space $(X,d,\mathfrak{m})$ , there exists $k\leq N$ s.t. $\mathfrak{m}(X-\mathcal{R}^{k}(X))=0$ [5]. We call this $k$ the rectifiable dimension of $(X,d,\mathfrak{m})$ .

Then we review the theory of almost splitting for RCD spaces, following the notations in [4].

Definition 2.3.

Let $(X,d,\mathfrak{m})$ be a RCD $(K,N)$ space for some $K\in\mathbb{R}$ and $1\leq N<\infty$ . Let $p\in X$ and $s>0$ . A map $u:B_{2s}(p)\to\mathbb{R}^{k}$ is a $(k,\delta)$ -splitting map if it belongs to the domain of the local Laplacian on $B_{2s}(p)$ , and

|\nabla u|_{|B_{s}(p)}\leq C(N),

\sum_{a,b=1}^{k}\fint_{B_{s}(p))}|\langle\nabla u^{a},\nabla u^{b}\rangle-\delta_{a}^{b}|\leq\delta^{2},

\sum_{a=1}^{k}\fint_{B_{s}(p))}s^{2}|\mathrm{Hess}(u^{a})|^{2}\leq\delta^{2}.

Such a map $u$ is sometimes referred as almost $k$ -splitting if it is $(k,\delta)$ -splitting for some sufficiently small $\delta>0$ . In the literature, it is often assumed that an almost $k$ -splitting map is harmonic. However, it is convenient to drop the harmonicity assumption in this paper. By Lemma 4.4 in [22], if we further assume that $\fint_{B_{2s}(p)}|\nabla\Delta u|\leq\delta$ , then $|\nabla u|_{|B_{s}(p)}\leq 1+\Phi(\delta|N,k)$ where $\Phi(\delta|N,k)\to 0$ as $\delta\to 0$ .

It is a classical result that the existence of an almost splitting function is equivalent to pmGH closeness to a space that splits off a Euclidean factor, see [4, 7, 8, 13].

We will use the fact that isometry group of a $\mathrm{RCD}(K,N)$ space is a Lie group [39, 21].

By [14], the generalized Margulis lemma holds for any RCD $(K,N)$ space, see also [28] for the manifolds with Ricci curvature bounded from below.

Theorem 2.4 (Generalized Margulis lemma, [14]).

For any $K\in\mathbb{R}$ and $N\geq 1$ , there exists $C$ and $\epsilon_{0}>0$ such that for any RCD $(K,N)$ space $(X,d,\mathfrak{m},p)$ with rectifiable dimension $k$ , the image of the natural homomorphism

\pi_{1}(B_{\epsilon_{0}}(p),p)\to\pi_{1}(B_{1}(p),p)

contains a normal nilpotent subgroup of index $\leq C$ . Moreover, this nilpotent subgroup has a nilpotent basis of length at most $k$ .

We now recall the volume convergence theorem for non-collapsed RCD $(K,N)$ spaces.

Theorem 2.5 (Volume convergence, [33]).

Assume that $(X_{i},d_{i},\mathcal{H}^{N},p_{i})$ is a sequence of non-collapsed RCD $(K,N)$ spaces which pointed measured GH converge to $(X,d,\mathfrak{m},p)$ . If $\mathcal{H}^{N}(B_{1}(p_{i}))\geq v>0$ , then $\mathfrak{m}=\mathcal{H}^{N}$ and $\forall r>0$ , $\mathcal{H}^{N}(B_{r}(p_{i}))$ converges to $\mathcal{H}^{N}(B_{r}(p))$ .

We next state the transformation theorem in [4], see also [12, 13, 22].

Theorem 2.6 (Transformation, [4]).

Let $1\leq N<\infty$ . For any $\delta>0$ there exists $\epsilon(N,\delta)>0$ such that for any $\epsilon<\epsilon(N,\delta)$ and any $x$ in a RCD $(-\epsilon^{2}(N-1),N)$ space $(X,d,\mathcal{H}^{N})$ , the following holds. If $B_{s}(x)$ is an $(N,\epsilon^{2})$ -symmetric ball for any $r_{0}\leq s\leq 1$ and $u:B_{2}(x)\to\mathbb{R}^{N}$ is a $(N,\epsilon)$ -splitting map, then for each scale $r_{0}\leq s\leq 1$ there exists an $N\times N$ lower triangular matrix $T_{s}$ such that
(1) $T_{s}u:B_{s}(x)\to\mathbb{R}^{N}$ is an $(N,\delta)$ -splitting map on $B_{s}(x)$ ;
(2) $\fint_{B_{s}(x)}\nabla(T_{s}u)^{a}\cdot\nabla(T_{s}u)^{b}d\mathcal{H}^{N}=\delta_{a}^{b}$ ;
(3) $|T_{s}\circ T_{2s}^{-1}-\mathrm{Id}|\leq\delta$ .

An application of the transformation theorem is to construct a biHöder map [12, 22].

Theorem 2.7 (Canonical Reifenberg method [12, 22]).

Assume that $(X,d,\mathcal{H}^{n},p)$ is a RCD $(-\epsilon^{2}(N-1),N-1)$ space and $u:B_{2}(p)\to\mathbb{R}^{N}$ is a harmonic $(N,\epsilon)$ -splitting map. Then for any $x,y\in B_{1}(p)$ we have

(1-\Phi(\epsilon|n,v))d(x,y)^{1+\Phi(\epsilon|n,v)}\leq d(f(x),f(y))\leq(1+\Phi(\epsilon|n,v))d(x,y).

Moreover, if $X$ is a smooth $N$ -manifold with Ric $\geq-\epsilon^{2}(N-1)$ , then for any $x\in B_{1}(p)$ , $du:T_{x}\to\mathbb{R}^{N}$ is nondegenerate.

In Theorem 2.7, the harmonicity condition on $u$ can be replaced by the condition $\fint_{B_{s}(p)}|\nabla\Delta u|\leq\epsilon$ [22].

2.3. The universal covers and local relatives covers of RCD $(K,N)$ spaces

For any RCD $(K,N)$ space $(X,d,\mathfrak{m})$ , the universal cover $\widetilde{X}$ exists and is a RCD $(K,N)$ space with induced metric and measure [31]. $X$ is semi-locally simply connected [41, 42], thus the fundamental group is isomorphic to the deck transformation group of $\widetilde{X}$ .

We first consider a compact RCD $(K,N)$ space $(X,d,\mathfrak{m},p)$ with diameter less than $D$ . Let $(\widetilde{X},\tilde{p})$ be the universal cover and $G$ be the fundamental group of $X$ . Define the set

G(\tilde{p},20D)=\{g\in G|d(\tilde{p},g\tilde{p})\leq 20D\}.

We shall review that $\bar{B}_{10D}(\tilde{p})$ and $G(\tilde{p},20D)$ determine $\widetilde{X}$ and $G$ [19, 37, 40].

$G(\tilde{p},20D)$ is a pseudo-group, i.e. for some $g_{1},g_{2}\in G(\tilde{p},20D)$ , $g_{1}g_{2}$ is not defined within $G(\tilde{p},20D)$ . To handle this, we define the groupfication $\hat{G}$ of $G(\tilde{p},20D)$ as follows. Let $F$ be the free group generated by elements $e_{g}$ for each $g\in G(\tilde{p},20D)$ . We may quotient $F$ by the normal subgroup generated by all elements of the form $e_{g_{1}}e_{g_{2}}e_{g_{1}g_{2}}^{-1}$ , where $g_{1},g_{2}\in G(\tilde{p},20D)$ with $g_{1}g_{2}\in G(\tilde{p},20D)$ . The quotient group is denoted $\hat{G}$ .

There is a natural (pseudo-group) homomorphism

i:G(\tilde{p},20D)\to\hat{G},\ i(g)=[e_{g}],

where $[e_{g}]$ is the quotient image of $e_{g}$ . Define

\pi:\hat{G}\to G,\ \pi([e_{g_{1}}e_{g_{2}}...e_{g_{k}}])=g_{1}g_{2}...g_{k}.

Then $\pi\circ i$ is the identity map on $G(\tilde{p},20D)$ , thus $i$ is injective. Since $G(\tilde{p},20D)$ generates $G$ , $\pi$ is surjective.

Now we can glue a space by equivalence relation,

\hat{G}\times_{G(\tilde{p},20D)}\bar{B}_{10D}(\tilde{p})=\hat{G}\times\bar{B}_{10D}(\tilde{p})/\sim,

where the equivalence relation is given by $(\hat{g}i(g),x)\sim(\hat{g},gx)$ for all $g\in G(\tilde{p},20D),\hat{g}\in\hat{G},x\in\bar{B}_{10D}(\tilde{p})$ with $gx\in\bar{B}_{10D}(\tilde{p})$ . Endow $\hat{G}\times_{G(\tilde{p},20D)}\bar{B}_{10D}(\tilde{p})$ with the induced length metric. Then $\hat{G}\times_{G(\tilde{p},20D)}\bar{B}_{10D}(\tilde{p})$ is a covering space of $\tilde{X}$ with deck transformation group $\ker(\pi)$ . $\tilde{X}$ is simply connected, thus we obtain the following result.

Theorem 2.8.

([19, 37, 40]) $\hat{G}$ is isomorphic to $G$ and $\hat{G}\times_{G(\tilde{p},20D)}\bar{B}_{10D}(\tilde{p})$ with induced length metric is isometric to $\widetilde{X}$ .

We next consider the local relative covers of a RCD $(K,N)$ space $(X,d,\mathfrak{m})$ . By [38], there exists a sequence of compact $n$ -manifolds $(M_{i},p_{i})$ with a uniform lower sectional curvature bound, so that the local universal cover of $\widetilde{B_{1}(p_{i})}$ admits no converging subsequence in the pointed Gromov-Hausdorff sense. To address this, we refer to a precompactness theorem for relative covers of open balls by Xu [43].

For any $p\in X$ , $r_{2}>r_{1}>0$ , define local relative over $(\widetilde{B}(p,r_{1},r_{2}),\tilde{p})$ as the a connected component of the pre-image of $B_{r_{1}}(p)$ in the universal cover of $B_{r_{2}}(p)$ , where $\tilde{p}$ is a pre-image point of $p$ .

Theorem 2.9 (Precompactness of relative covers, [43]).

$(\widetilde{B}(p,r_{1},r_{2}),\tilde{p})$ equipped with its length metric and measure, is globally $A_{K,N,r_{1},r_{2}}(r)$ -doubling, that is, there exists a positive non-decreasing function $A_{K,N,r_{1},r_{2}}(r)$ such that

0<\mathfrak{m}(B_{r}(x))\leq A_{K,N,r_{1},r_{2}}(r)\mathfrak{m}(B_{\frac{r}{2}}(x)),\text{ \ for any \ }x\in(\widetilde{B}(p,r_{1},r_{2}),\tilde{p}).

In particular, let $G$ be the image of $\pi_{1}(B_{r_{1}}(p),p)\to\pi_{1}(B_{r_{2}}(p),p)$ , then the family consisting of all such triples $(\widetilde{B}(p,r_{1},r_{2}),\tilde{p},G)$ is precompact in the pointed equivariant Gromov-Hausdorff topology.

Remark 2.10.

In Theorem 2.9, the quotient space $\widetilde{B}(p,r_{1},r_{2})/G$ is isometric to $B_{r_{1}}(p)$ with the length metric on itself. The length metric on $B_{r_{1}}(p)$ may differ from the original metric $d$ . However, they coincide on $B_{\frac{r_{1}}{3}}(p)$ . To simplify the notation, we always assume that the length metric on $B_{r_{1}}(p)$ is the original metric, otherwise we consider $B_{\frac{r_{1}}{3}}(p)$ with length metric on $B_{r_{1}}(p)$ .

We state some technical lemmas.

Lemma 2.11.

Let $G$ be a nilpotent Lie group and $G_{0}$ be the identity component. Then for any compact subgroup $K\subset G$ , the commutator $[K,G_{0}]$ is trivial.

Lemma 2.12 (Covering lemma, [28]).

There exists $C(N)$ such that the following holds. Let $(X,d,\mathfrak{m},p)$ be a RCD $(-(N-1),N)$ space and $f:X\to\mathbb{R}$ is a non-negative function. Let $\pi:\widetilde{X}\to X$ be the universal cover of $X$ and $\tilde{p}\in\widetilde{X}$ is a lift of $p$ . Let $\tilde{f}=f\circ\pi$ , then

C^{-1}(N)\fint_{B_{\frac{1}{3}}(p)}f\leq\fint_{B_{1}(\tilde{p})}\tilde{f}\leq C(N)\fint_{B_{1}(p)}f.

Lemma 2.13 (Gap lemma, [28]).

Assume that $(X_{i},p_{i},G_{i})$ is a sequence of length metric space and

(X_{i},p_{i},G_{i})\overset{eGH}{\longrightarrow}(X,p,G).

Assume that there exists $b>a>0$ such that $\langle G(p,r)\rangle$ is the same group for any $r\in(a,b)$ . Then there exists $\epsilon_{i}\to 0$ , so that for any sufficiently large $i$ , $\langle G_{i}(p_{i},r)\rangle$ is the same group for any $r\in(a+\epsilon_{i},b-\epsilon_{i})$ .

2.4. Approximate groups and almost homogeneous spaces

The references of this subsection are [3, 44].

Definition 2.14.

A (symmetric) local group $G$ is a topological space with the identity element $e\in G$ , together with a global inverse map $()^{-1}:G\to G$ and a partially defined product map $\cdot:\Omega\to G$ , satisfying the following axioms:
(1) $\Omega$ is an open neighborhood of $(G\times\{e\}\cup(\{e\}\times G)$ in $G\times G$ .
(2) The map $()^{-1}:x\to x^{-1}$ and $\cdot:(x,y)\to xy$ are continuous.
(3) If $g,h,k\in G$ s.t. that $(gh)k$ and $g(hk)$ are well-defined, then $(gh)k=g(hk)$ .
(4) For any $g\in G$ , $eg=ge=g$ .
(5) For any $g\in G$ , $gg^{-1}$ and $g^{-1}g$ are well-defined and equal to $e$ .

In particular, if $\Omega=G\times G$ , we call $G$ a global group or a topological group.

Definition 2.15.

Let $G$ be a local group and $g_{1},g_{2},...,g_{m}\in G$ . We say that the product $g_{1}g_{2}...g_{m}$ is well-defined, if for each $1\leq i\leq j\leq m$ we can find $g_{[i,j]}\in G$ s.t. $g_{[k,k]}=g_{k}$ for any $1\leq k\leq m$ and $g_{[i,j]}g_{[j+1,k]}$ is well-defined and equal to $g_{[i,k]}$ for any $1\leq i\leq j<k\leq m$ .

For sets $A_{1},A_{2},...,A_{m}\subset G$ , we say the product $A_{1}A_{2}...A_{m}$ is well-defined if for any choices of $g_{j}\in A_{j}$ , $g_{1}g_{2}...g_{m}$ is well-defined.

Definition 2.16.

$A\subset G$ is called a multiplicative set if it is symmetric $A=A^{-1}$ , and $A^{200}$ is well-defined.

Definition 2.17.

Let $A$ be a finite symmetric subset of a multiplicative set and $C\in\mathbb{N}$ , $A$ is called a $C$ -approximate group if $A^{2}$ can be covered by $C$ left translate of $A$ .

Definition 2.18.

Let $A$ be a $C$ -approximate group. We call $A$ a strong $C$ -approximate group if there is a symmetric set $S\subset A$ so that
(1) $(\{asa^{-1}|s\in S,a\in A\})^{10^{3}C^{3}}\subset A$ ;
(2) if $g,g^{2},...,g^{1000}\in A^{100}$ , then $g\in A$ ;
(3) if $g,g^{2},...,g^{10^{3}C^{3}}\in A^{100}$ , then $g\in S$ .

Consider a multiplicative set $A\subset G$ . For any $g\in G$ , define the escape norm as

||g||_{A}=:\inf\{\frac{1}{m+1}|e,g,g^{2},...,g^{m}\in A\}.

Theorem 2.19 (escape norm estimate, [3]).

For each $C>0$ , there is $M>0$ s.t. if $A$ is a strong $C$ -approximate group and $g_{1},g_{2},...,g_{k}\in A^{10}$ , then
(1) $||g_{1}g_{2}...g_{k}||_{A}\leq M\sum_{j=1}^{k}||g_{j}||_{A}$ .
(2) $||g_{2}g_{1}g_{2}^{-1}||_{A}\leq 10^{3}||g_{1}||_{A}$ .
(3) $||[g_{1},g_{2}]||_{A}\leq M||g_{1}||_{A}||g_{2}||_{A}$ .

Remark 2.20.

Due to (1) and (2), the set $w=\{g\in A|||g||_{A}=0\}$ is a normal subgroup of $A$ . We call that $A$ contains no small subgroup if $w$ is trivial. For any general strong approximate group $A$ , $A/w$ is a strong approximate group with no small subgroup.

Readers may compare (3) in Theorem 2.19 with the holonomy estimates in Gromov’s argument for almost flat manifolds, see [6] Chapter 2.

Definition 2.21.

(nilprogression) Let $G$ be a local group, $u_{1},u_{2},...,u_{r}\in G$ and $C_{1},C_{2},...,C_{r}\in\mathbb{R}^{+}$ . The set $P(u_{1},...,u_{r};C_{1},...,C_{r})$ is defined as the the set of words in the $u_{i}$ ’s and their inverses such that the number of appearances of $u_{i}$ and $u_{i}^{-1}$ is not more than $C_{i}$ . We call $P(u_{1},...,u_{r};C_{1},...,C_{r})$ a nilprogession of rank $r$ if every word in it is well defined in $G$ . We say it a nilprogession in $C$ -normal form for some $C>0$ if it satisfies the following properties:
(1) For all $1\leq i\leq j\leq r$ , we have

[u_{i}^{\pm 1},u_{j}^{\pm 1}]\in P(u_{j+1},...,u_{r};\frac{CC_{j+1}}{C_{i}C_{j}},...,\frac{CC_{r}}{C_{i}C_{j}}).

(2) The expression $u_{1}^{n_{1}}...u_{r}^{n_{r}}$ represent different elements in $G$ for $|n_{1}|\leq\frac{C_{1}}{C},...,|n_{r}|\leq\frac{C_{r}}{C}$ .
(3) $\frac{1}{C}(2\lfloor C_{1}\rfloor+1)...(2\lfloor C_{r}\rfloor+1)\leq|P|\leq C(2\lfloor C_{1}\rfloor+1)...(2\lfloor C_{r}\rfloor+1)$ .

For a nilprogression $P(u_{1},...,u_{r};C_{1},...,C_{r})$ in $C$ -normal form and $\epsilon\in(0,1)$ , define $\epsilon P=P(u_{1},...,u_{r};\epsilon C_{1},...,\epsilon C_{r})$ . Define the thickness of $P$ as the minimum of $C_{1},...,C_{r}$ and we denote it by thick $(P)$ . The set $\{u_{1}^{n_{1}}...u_{r}^{n_{r}}||n_{1}|\leq C_{1}/C,...,|n_{r}|\leq C_{r}/C\}$ is called the grid part of $P$ and is denoted by $G(P)$ .

Definition 2.22.

Let $P(u_{1},...,u_{r};C_{1},...,C_{r})$ be a nilprogression in $C$ -normal form with thick $(P)>C$ . Set $\Gamma_{P}$ to be the abstract group generated by $\gamma_{1},...,\gamma_{r}$ with the relation $[\gamma_{j},\gamma_{k}]=\gamma_{k+1}^{\beta_{j,k}^{k+1}}...\gamma_{r}^{\beta_{j,k}^{r}}$ whenever $j<k$ , where $[u_{j},u_{k}]=u_{k+1}^{\beta_{j,k}^{k+1}}...u_{r}^{\beta_{j,k}^{r}}$ and $|\beta_{j,k}^{l}|\leq\frac{CN_{l}}{N_{j}N_{k}}$ . We say that $P$ is good if each element of $\Gamma_{P}$ has a unique expression of the form $\gamma_{1}^{n_{1}}...\gamma_{r}^{n_{r}}$ with $n_{1},...,n_{r}\in\mathbb{Z}$ .

Theorem 2.23 ([3, 44]).

For each $r\in\mathbb{N},C>0$ , there is $\epsilon>0$ so that the following holds. Let $P(u_{1},...,u_{r};C_{1},...,C_{r})$ be a nilprogression in $C$ -normal form. if thick $(P)$ is large enough depending on $r$ and $C$ , then $P$ is good and the map $u_{j}\to\gamma_{j}$ extends to a product preserving embedding from $G(\epsilon P)$ to $\Gamma_{P}$ . And $\Gamma_{P}$ is isomorphic to the lattice in a $r$ -dim simply connected nilpotent Lie group $\mathcal{N}$ .

Remark 2.24.

During the proof of Theorem A and B, we shall use nilprogressions $P$ satisfying Theorem 2.23. We may simply identify the grid part $G(\epsilon P)$ and $P$ . Then we simply call $\Gamma_{P}$ in Theorem 2.23 the groupfication of $P$ .

In [44], Zamora used the structure of approximates groups to study the limit of almost homogeneous spaces. A sequence of geodesic metric spaces $Z_{i}$ is called almost homogeneous if there are discrete isometric group actions $G_{i}$ on $Z_{i}$ with diam $(Z_{i}/G_{i})\to 0$ . Now we assume that $(Z_{i},p_{i},G_{i})\overset{eGH}{\longrightarrow}(Z,p,G)$ . If we further assume that $Z$ is semi-locally simply connected, Zamora proved that $G$ is a Lie group.

(2.1)

\displaystyle\begin{CD}(Z_{i},p_{i},G_{i})@>{eGH}>{}>(Z,p,G)\\ @V{}V{\pi}V@V{}V{\pi}V\\ M_{i}^{\prime}@>{GH}>{}>\mathrm{pt}\end{CD}

Remark 2.25.

If we further assume that $Z_{i}$ is a RCD $(-(N-1),N)$ space, then the limit $Z$ is also a RCD $(-(N-1),N)$ space with a limit measure. Therefore $G$ is a Lie group and $Z$ must be semi-locally simply connected. We may always assume that $Z_{i}$ and $Z$ in 2.1 are RCD spaces.

Assume dim $(G)=r$ . A small neighborhood of the identity $e\in G$ is a strong $C$ approximate group for some $C>0$ . Thus for any small $\delta$ ,

G_{i}(\delta)=\{g\in G_{i}|d(gx,x)\leq\delta,\forall x\in\bar{B}_{\frac{1}{\delta}}(p_{i})\}

is a strong $C$ approximate group.

We say that $G_{i}$ has no small subgroup if there is no non-trivial subgroup of $G_{i}$ converging to the identity $e\in G$ as $i\to\infty$ . The next Theorem states that if $G_{i}$ has no small subgroup and equivariantly converges to a Lie group, then $G_{i}$ contains a large nilprogression which includes a neighborhood of the identity element $e\in G_{i}$ .

Theorem 2.26 ([3, 44]).

Assume that $G_{i}$ in 2.1 contains no small small subgroup, then for any $\epsilon>0$ sufficiently small, there exists $\delta>0$ independent of the choice of $i$ , such that $G_{i}(\epsilon)$ contains a nilprogession $P_{i}(u_{1},...,u_{r};C_{1},...,C_{r})$ in $C$ -normal form for some constant $C>0$ and $G_{i}(\delta)\subset G(P_{i})$ .

Remark 2.27.

The original statement of Theorem 2.26 in [3, 44] is purely algebraic using ultralimits. It is convenient in this paper to state it geometrically using equivariant convergence.

We first clarify the notations of exponential maps. If $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and a left-invariant metric, we denote by

\mathrm{exp}_{G}:\mathfrak{g}\to G

as the Lie group exponential map. For the identity element $e$ , define the Riemannian exponential map at $e$ :

\mathrm{exp}_{e}:T_{e}G\to G,

where $T_{e}G$ is the tangent space to $G$ at the identity.

We briefly recall how to construct the nilprogression in Theorem 2.26. Choose $\epsilon$ small enough so that the set

G(10\epsilon)=\{g\in G|d(gx,x)\leq 10\epsilon,\forall x\in\bar{B}_{\frac{1}{10\epsilon}}(p)\}

is connected and $\exp^{-1}_{G}$ from $G(10\epsilon)$ to the Lie algebra of $G$ is diffeomorphic.

Since $G_{i}(\epsilon)$ is a strong approximate group with no small subgroup, in particular, the escape norm is always non-zero. we find the element $u_{1}$ with the smallest escape norm. We may assume the diameter of $Z_{i}$ is small enough so that there are generators of $G_{i}$ with norm $<1/M$ , where $M$ is a constant obtained from Theorem 2.19. Then by (3) in Theorem 2.19, the commutator $[u_{1},g]$ is trivial for any $g$ in the chosen generators; otherwise we get a non-trivial element whose escape norm is strictly less than $u_{1}$ ’s, a contradiction. In particular, $u_{1}$ must lie in the center of $G_{i}(\delta)$ . The group generated by $u_{1}$ , $\langle u_{1}\rangle$ , converges to an one-parameter subgroup in the center of $G$ . By taking quotient groups and applying an induction argument, we can construct the nilprogression $P_{i}$ .

An important observation from the above construction is that the nilpotent structure of $P_{i}$ is determined by the escape norm of $G_{i}(\epsilon)$ . We shall use this observation to prove that structure group is affine in Theorem B.

Next recall the structure theory for nilpotent Lie groups and their Lie algebras.

Definition 2.28.

Let $\mathfrak{g}$ be a nilpotent Lie algebra. We say that an ordered basis $\{v_{1},...,v_{r}\}$ of $\mathfrak{g}$ is a strong Malcev basis if for any $1\leq k\leq r$ , the vector subspace $J_{k}$ generated by $\{v_{k+1},...,v_{r}\}$ is an ideal, and $v_{k}+J_{k}$ is in the center of $\mathfrak{g}/J_{k}$ .

Theorem 2.29.

Let $\mathcal{N}$ be a $r$ -dim simply connected nilpotent Lie group and $\mathfrak{g}$ be its Lie algebra with a strong Malcev basis $\{v_{1},...,v_{r}\}$ . Then:
(1) $\mathrm{exp}_{\mathcal{N}}:\mathfrak{g}\to\mathcal{N}$ is a diffeomorphism;
(2) $\phi:\mathbb{R}^{r}\to\mathcal{N}$ given by $\phi(x_{1},...,x_{r})=\mathrm{exp}_{\mathcal{N}}(x_{1}v_{1})...\mathrm{exp}_{\mathcal{N}}(x_{r}v_{r})$ is a diffeomorphism;
(3) if we identify $\mathfrak{g}$ with $\mathbb{R}^{r}$ by the given basis, then $\mathrm{exp}_{\mathcal{N}}^{-1}\circ\phi:\mathfrak{g}\to\mathfrak{g}$ and $\phi^{-1}\circ\mathrm{exp}_{\mathcal{N}_{i}}:\mathfrak{g}\to\mathfrak{g}$ are polynomials of degree bounded by a number depending only on $r$ .

In the diagram 2.1, for sufficiently large $i$ , by Theorem 2.23 and 2.26, the (grid part of) nilprogression $P_{i}$ can be identified as a generating set of a lattice in simply connected nilpotent group $\mathcal{N}_{i}$ . Let $g_{i,j}=u_{j}^{\lfloor C_{j}/C\rfloor}$ and $v_{i,j}$ in the Lie algebra $\mathfrak{g}_{i}$ of $\mathcal{N}_{i}$ such that

\mathrm{exp}_{\mathcal{N}_{i}}(v_{i,j})=g_{i,j},\ 1\leq j\leq r.

Then $\{v_{i,j},1\leq j\leq r\}$ is a strong Malcev basis of $\mathfrak{g}_{i}$ . For any $j$ , passing to a subseqeunce if necessary, assume $g_{i,j}\to g_{j}\in G$ and choose $v_{j}$ in the Lie algebra $\mathfrak{g}$ of $G$ such that $\mathrm{exp}_{G}(v_{j})=g_{j}$ .

For any fixed $i$ , since $\{v_{i,j},1\leq j\leq r\}$ is a strong Malcev basis of $\mathfrak{g}_{i}$ , then $\phi_{i}:\mathbb{R}^{r}\to\mathcal{N}_{i}$ , as in Theorem 2.29, is a diffeomorphism. Now we identity $(x_{1},x_{2},...,x_{r})\in\mathbb{R}^{r}$ as $\sum_{j=1}^{r}x_{j}v_{i,j}\in\mathfrak{g}_{i}$ , and define

Q_{i}:\mathbb{R}^{r}\times\mathbb{R}^{r}\to\mathbb{R}^{r},\ Q_{i}(x,y)=\phi_{i}^{-1}(\phi_{i}(x)\phi_{i}(y)).

Similarly define $Q$ for $G$ and $\{v_{j},1\leq j\leq r\}$ . Roughly speaking, $Q_{i}$ and $Q$ decide Lie algebra structure of $\mathcal{N}_{i}$ and $G$ respectively.

Theorem 2.30 (Lie algebra structure convergence, [44]).

For sufficiently large $i$ , $Q_{i}$ and $Q$ are all polynomials of degree $\leq d(r)$ and coefficients of $Q_{i}$ converge to corresponding ones of $Q$ .

3. Constructing a GHA map which is locally almost splitting

In this section, we want to generalize main results in [23] from the smooth case to a weaker version in the RCD case.

Theorem 3.1 (smooth fibration, [23]).

Given $n,v>0$ , there exists $\epsilon(n,v)>0$ and $C(n)>0$ so that consider a compact $n$ -manifold $M$ and a $k$ -manifold $K$ satisfying:
$\text{Ric}_{M}\geq-(n-1)$ , $(1,v)$ -bound covering geometry holds on $M$ , $|\text{sec}_{K}|\leq 1$ , $\text{inj}_{K}\geq 1$ , $d_{GH}(M,K)<\epsilon<\epsilon(n,v)$ .
Then there is a smooth fiber bundle map $f:M\to K$ which is a $\Phi(\epsilon|n,v)$ -GHA, where $\Phi(\epsilon|n,v)\to 0$ as $\epsilon\to 0$ .

Assume a group $G$ isometrically acts two metric spaces $X_{1}$ and $X_{2}$ separately, we call a map $h:X_{1}\to X_{2}$ $\epsilon$ -almost $G$ -equivariant if $d(h(gx),gh(x))<\epsilon$ for any $x\in X_{1},g\in G$ .

Theorem 3.2 (stability for compact group actions, [23]).

There exists $\epsilon(n)>0$ so that the following holds for any $\epsilon<\epsilon(n)$ . Assume that $M$ and $K$ are compact $n$ -manifolds so that $|\mathrm{sec}_{K}|\leq 1$ , $\mathrm{inj}_{K}\geq 1$ , $\mathrm{Ric}_{M}\geq-(n-1)$ . The group $G$ acts isometrically on $M$ and $K$ separately and there is $\epsilon$ -GHA $h:M\to K$ which is $\epsilon$ -almost $G$ -equivariant. Then there exists a $G$ -equivariant diffeomorphism $f:M\to K$ , that is $f(gx)=gf(x)$ for any $x\in M$ and $g\in G$ , which is a $\Phi(\epsilon|n)$ -GHA.

We briefly recall the proof of Theorem 3.1 and 3.2. In Theorem 3.1, the injective radius of $K$ is at least $1$ . Then locally we can identify a small ball in $K$ as an open subset in the tangent space of $K$ at some point. By our assumption that $M$ is GH close to a manifold $K$ , locally we can construct almost $k$ -splitting maps from a small open neighborhood in $M$ to the tangent space of $K$ at some point.

To construct a globally-defined map $f:M\to K$ , we can glue and average these local almost splitting maps using some cut-off functions and the center of mass technique. Then we have a smooth GHA $f:M\to K$ . Then Huang showed that, at any point $p\in M$ , $df$ is the same as the differential of a local almost $k$ -splitting function, which is constructed by the linear average. Then Huang proved that the differential of any almost $k$ -splitting map is non-degenerate under $(1,v)$ -bound covering condition. Thus $df$ is non-degenerate and $f$ is a fibration map by the implicit function theorem.

The proof of Theorem 3.2 follows a similar approach. Huang construct a $G$ -equivariant map using the center of mass and applies the canonical Reifenberg method to show that the map is a diffeomorphism.

In the non-smooth RCD case, we have no implicit function theorem. However, we can prove the following two theorems using ideas from [23]. For any metric space $(X,d)$ and $p\in X$ , $r>0$ , we use $rX$ or $(X,rd)$ for the rescaled metric on $X$ . Then $rB_{\frac{1}{r}}(p)$ is actually a unit ball in $rX$ .

Theorem 3.3 (Almost $k$ -splitting).

Assume that a sequence of compact RCD $(-(N-1),N)$ spaces $(X_{i},d_{i},\mathfrak{m})$ converges to a smooth compact $k$ -manifold $(K,g)$ in the Gromov-Hausdorff sense. Then for sufficiently large $i$ , there is a continuous GHA $f_{i}:X_{i}\to K$ which is local almost $k$ -splitting, i.e., for any $\delta>0$ , $p_{i}\in X_{i}$ close to $p\in K$ , and $i$ large enough,

\frac{1}{\delta}B_{\sqrt{\delta}}(p_{i})\overset{f_{i}}{\longrightarrow}\frac{1}{\delta}B_{\sqrt{\delta}}(p)\overset{exp_{p}^{-1}}{\longrightarrow}T_{p}K=\mathbb{R}^{k}

is a $(k,\delta)$ -splitting $\delta$ -GHA.

Proof.

Take a small number $\epsilon<1$ . We may assume that the injective radius of $K$ is at least $\frac{10}{\epsilon}$ and $X_{i}$ is RCD $(-(N-1)\epsilon^{2},N)$ . We also assume that for any $p\in K$ , $B_{\frac{10}{\epsilon}}(p)$ is $\epsilon^{2}$ - $C^{4}$ -close, by $\mathrm{exp}_{p}^{-1}$ , to its pre-image in $T_{p}K$ with the flat metric. Otherwise we can consider $(K,rg)$ for sufficiently large $r$ , then $(X_{i},rd_{i})$ will still converges to $(K,rg)$ .

Let $\Phi(\epsilon|k,N)$ denote a function which converges to $0$ as $\epsilon\to 0$ , for fixed $k,N$ . The value of $\Phi(\epsilon|k,N)$ may vary depending on the specific case. Take any large $i$ such that $d_{GH}(X_{i},K)<\epsilon$ . Our goal is to construct a $\Phi(\epsilon|k,N)$ -GHA $f_{i}:X_{i}\to K$ which is $(k,\Phi(\epsilon|k,N))$ -splitting on any $\frac{1}{\epsilon}$ ball. Once this is established, we just take $\epsilon$ small enough so that $\Phi(\epsilon|k,N)<\delta$ , thereby completing the proof.

Let $\{p^{j}\}_{j=1,2,...J}$ be a $\frac{1}{\epsilon}$ -net in $K$ and find $p_{i}^{j}\in X_{i}$ $\epsilon$ -close to $p^{j}\in K$ for each $j$ . $\Lambda_{j}$ is a cut-off function on $K$ such that $\Lambda_{j}(B_{\frac{1}{\epsilon}}(p^{j}))=1$ and $\mathrm{supp}(\Lambda_{j})\subset B_{\frac{2}{\epsilon}}(p^{j})$ . We may assume $|\nabla^{l}\Lambda_{j}|\leq\Phi(\epsilon|k,N)$ , $l=1,2,3$ . Let $B_{ij}=B_{\frac{2}{\epsilon}}(p_{i}^{j})$ , $B_{j}=B_{\frac{2}{\epsilon}}(p^{j})$ , $B_{j}^{-1}$ be the pre-image of $B_{j}$ in $T_{p^{j}}K$ with the flat metric.

$B_{ij}$ is $\epsilon$ -GH close to $B_{j}$ , $B_{j}$ is $\epsilon^{2}$ - $C^{4}$ -close to the $B^{-1}_{j}$ with the flat metric. Take a smaller radius if necessary, we can construct a $\Phi(\epsilon|k,N)$ -GHA

h_{j}:B_{ij}\longrightarrow B_{j},

such that $\mathrm{exp}_{p^{j}}^{-1}\circ h_{j}:B_{ij}\to B_{j}^{-1}\subset\mathbb{R}^{k}$ is a harmonic $(k,\Phi(\epsilon|k,N))$ -splitting map.

Take the energy function $E:X_{i}\times K\to\mathbb{R}$ as follows,

E(x,y)=\frac{\sum_{j=1}^{J}\Lambda_{j}(h_{j}(x))d(h_{j}(x),y)^{2}}{\sum_{j=1}^{J}\Lambda_{j}(h_{j}(x))}.

Since $\mathrm{supp}(\Lambda_{j})$ is contained in the image of $h_{j}$ , $\Lambda_{j}(h_{j}(x))$ is well-defined for any $x\in X_{i}$ by a $0$ extension outside of the support. Since $h_{j}$ is a GHA, the image of all $\{h_{j}(x)\}_{j=1,2,...,J}$ (if defined) is contained in a $\Phi(\epsilon|k,N)$ -ball for any fixed $x\in X_{i}$ . Let $y^{\prime}\in K$ be a point close to $x$ , then $E(x,\cdot)$ is strictly convex in $B_{\frac{1}{\epsilon}}(y^{\prime})$ and achieve a global minimum at some $y\in B_{\frac{1}{\epsilon}}(y^{\prime})$ , which is the center of mass with respect to $E$ . Define $f_{i}(x)=y$ , then $f_{i}$ is a $\Phi(\epsilon|k,N)$ -GHA.

We next show that $f_{i}$ is $(k,\Phi(\epsilon|k,N))$ -splitting map on any $\frac{1}{\epsilon}$ -ball. For any $x_{0}\in X_{i}$ , take $y_{0}=f(x_{0})$ . There exists at most $C(N)$ many points $p^{j}$ in the net contained in $B_{\frac{4}{\epsilon}}(y_{0})$ , saying $j_{1},j_{2},j_{3},...,j_{C}$ . Then the value of $f_{i}$ on $B_{\frac{1}{\epsilon}}(x_{0})$ only depends on $h_{j}$ and $\Lambda_{j}$ for $j=j_{1},j_{2},...,j_{C}$ .

Consider the energy function on the product space, $\tilde{E}:\prod_{l=1}^{C}B_{j_{l}}\times K\to\mathbb{R}$ ,

\tilde{E}(\prod_{l=1}^{C}y_{l},y)=\frac{\sum_{l=1}^{C}\Lambda_{j_{l}}(y_{l})d(y_{l},y)^{2}}{\sum_{l=1}^{C}\Lambda_{j_{l}}(y_{l})},\forall y\in K,y_{l}\in B_{j_{l}},l=1,2,...,C.

For any $\prod_{l=1}^{C}y_{l}\in\prod_{l=1}^{C}B_{j_{l}}$ , define $\tilde{h}(\prod_{l=1}^{C}y_{l})$ to be the center of mass with respect to $\tilde{E}$ . Then by the definition,

f_{i}(x)=\tilde{h}(h_{j_{1}}(x),h_{j_{2}}(x),...,h_{j_{C}}(x)),\ \forall x\in B_{\frac{1}{\epsilon}}(x_{0}).

Now consider the center of mass on the Euclidean space, which is a linear average. Define

\bar{h}:\prod_{l=1}^{C}B_{j_{l}}\to T_{y_{0}}K,\ \bar{h}(\prod_{l=1}^{C}y_{l})=\frac{\sum_{l=1}^{C}\Lambda_{j_{l}}(y_{l})\mathrm{exp}_{y_{0}}^{-1}(y_{l})}{\sum_{l=1}^{C}\Lambda_{j_{l}}(y_{l})}.

Then $\mathrm{exp}_{y_{0}}^{-1}\circ\tilde{h}$ is $\Phi(\epsilon|k,N)$ - $C^{3}$ close to $\bar{h}$ , since the metric on $B_{\frac{10}{\epsilon}}(y_{0})$ is $\epsilon^{2}$ - $C^{4}$ close the a $\frac{10}{\epsilon}$ -ball in $T_{y_{0}}K$ with the flat metric; the center of mass in flat $\mathbb{R}^{k}$ is the linear average.

For any $1\leq l\leq C$ , $\mathrm{exp}_{p^{j_{l}}}^{-1}\circ h_{j_{l}}$ is a harmonic $\Phi(\epsilon|k,N)$ -GHA and $\mathrm{exp}_{y_{0}}^{-1}\circ\mathrm{exp}_{p^{j_{l}}}$ is $\Phi(\epsilon|k,N)$ - $C^{3}$ -close to the a linear isometric action on $B_{\frac{3}{\epsilon}}(0^{k})$ by our assumption, thus

\mathrm{exp}_{y_{0}}^{-1}\circ h_{j_{l}}=\mathrm{exp}_{y_{0}}^{-1}\circ\mathrm{exp}_{p^{j_{l}}}\circ\mathrm{exp}_{p^{j_{l}}}^{-1}\circ h_{j_{l}}

is a $\Phi(\epsilon|k,N)$ -GHA with

|\Delta(\mathrm{exp}_{y_{0}}^{-1}\circ h_{j_{l}})|\leq\Phi(\epsilon|k,N),|\nabla\Delta(\mathrm{exp}_{y_{0}}^{-1}\circ h_{j_{l}})|\leq\Phi(\epsilon|k,N)

on $B_{\frac{1}{\epsilon}}(y_{0})$ .

Since $|\nabla^{l}\Lambda_{j}|\leq\Phi(\epsilon|k,N)$ , $l=1,2,3$ , the linear average

\bar{h}\circ(h_{j_{1}},h_{j_{2}},...,h_{j_{C}})=\frac{\sum_{l=1}^{C}\Lambda_{j_{l}}(h_{j_{l}}(x))\mathrm{exp}_{y_{0}}^{-1}(h_{j_{l}}(x))}{\sum_{l=1}^{C}\Lambda_{j_{l}}(h_{j_{l}}(x))}:B_{\frac{1}{\epsilon}}(x_{0})\to T_{y_{0}}K

is a $\Phi(\epsilon|k,N)$ -GHA with

|\Delta(\bar{h}(h_{j_{1}},h_{j_{2}},...,h_{j_{C}}))|\leq\Phi(\epsilon|k,N),|\nabla\Delta(\bar{h}(h_{j_{1}},h_{j_{2}},...,h_{j_{C}})|\leq\Phi(\epsilon|k,N)

on $B_{\frac{1}{\epsilon}}(y_{0})$ . In particular, $\bar{h}(h_{j_{1}},h_{j_{2}},...,h_{j_{C}})$ is a $(k,\Phi(\epsilon|k,N))$ -splitting map on $B_{\frac{1}{\epsilon}}(x_{0})$ .

Since $\mathrm{exp}_{y_{0}}^{-1}\circ\tilde{h}$ is $\Phi(\epsilon|k,N)$ - $C^{3}$ -close to $\bar{h}$ , $\mathrm{exp}_{y_{0}}^{-1}\circ f_{i}=\mathrm{exp}_{y_{0}}^{-1}\circ\tilde{h}(h_{j_{1}},h_{j_{2}},...,h_{j_{C}})$ is also a $(k,\Phi(\epsilon|k,N))$ -splitting map on $B_{\frac{1}{\epsilon}}(x_{0})$ . ∎

Remark 3.4.

We can also use the embedding argument in [22] to prove Theorem 3.1. By the Nash embedding theorem, we can isometrically embed $K$ into some $\mathbb{R}^{n}$ where $n$ only depends on $k$ . The embedding map is $\Phi=(\phi_{1},\phi_{2},...,\phi_{n}):K\hookrightarrow\mathbb{R}^{n}$ . Let $\pi$ be the projection map from a neighborhood of $K\subset\mathbb{R}^{n}$ to $K$ . By [22], if $i$ is large enough, we can construct $\Psi^{i}=(\psi_{1}^{i},\psi_{2}^{i},...,\psi_{n}^{i}):X_{i}\to\mathbb{R}^{n}$ so that $f=\Phi^{-1}\circ\pi\circ\Psi^{i}$ is a GHA and $\Delta\psi^{i}_{j}$ is $H^{1,2}$ close to $\Delta\phi_{i}$ for any $1\leq j\leq n$ . Then $f$ is locally almost $k$ -splitting if we sufficiently blow up the metric.

We next state a $G$ -stability result in the RCD setting.

Theorem 3.5 ( $G$ -equivariant).

There exists $\epsilon(N)$ such that for any $0<\epsilon\leq\epsilon(N)$ , the following holds. For any compact RCD $(-(N-1)\epsilon^{2},N)$ space $(X,d,\mathfrak{m})$ and $k$ -manifold $K$ satisfying that $\mathrm{inj}_{K}\geq\frac{10}{\epsilon}$ and for any $p\in K$ , $B_{\frac{10}{\epsilon}}(p)$ is $\epsilon^{2}$ - $C^{4}$ -close, by $\mathrm{exp}_{p}^{-1}$ , to its preimage in the tangent space $T_{p}K$ with the flat metric. Assume that a map $h:X\to K$ is an $\epsilon$ -GHA map on any $\frac{10}{\epsilon}$ -ball in $X$ , that is for any $x\in X$ , $h:B_{\frac{10}{\epsilon}}(x)\to K$ is an $\epsilon$ -GHA to its image. A finite group $G$ acts isometrically on $X$ and $K$ separately and $h$ is $\epsilon$ -almost $G$ -equivariant. Then there is a $G$ -equivariant map $f_{G}:X\to K$ , which is also a $\Phi(\epsilon|k,N)$ -GHA and locally $(k,\Phi(\epsilon|k,N))$ -splitting on any $\frac{1}{\epsilon}$ -ball.

Remark 3.6.

In Theorem 3.5, there is no control of the diameter of $X$ or the order $|G|$ . Instead we only need a global map $h$ and the gluing is local. If we further assume $k=N$ and $\epsilon$ small enough, then $f_{G}$ is biHölder on any $\frac{1}{\epsilon}$ -ball due to Theorem 2.7.

Proof.

We can use the same proof of Theorem 3.3 to construct a map $f:X\to K$ which is $(k,\Phi(\epsilon|k,N))$ -splitting and a $\Phi(\epsilon|k,N)$ -GHA on any $\frac{1}{\epsilon}$ -ball. Notice that the condition that $h:X\to K$ is an $\epsilon$ -GHA map on any $\frac{10}{\epsilon}$ -ball is enough for the construction in Theorem 3.3 as the gluing and averaging procedure is local.

Since $f(x)$ is close to $h(x)$ for any $x\in X$ , we have $d(f(gx),gf(x))\leq\Phi(\epsilon|k,N)$ for any $x\in X$ and $g\in G$ . In particular, for any $g\in G$ ,

gf(g^{-1}x):X\to K

is a $\Phi(\epsilon|k,N)$ -GHA.

Since $f$ is $(k,\Phi(\epsilon|k,N))$ -splitting on any $\frac{1}{\epsilon}$ -ball and $G$ -actions are isometric on $X$ and $K$ , thus for any $g\in G$ , the map $gf(g^{-1}x):X\to K$ is also $(k,\Phi(\epsilon|k,N))$ -splitting on any $\frac{1}{\epsilon}$ -ball.

Now we average $G$ actions by the center of mass. Take the energy function

E(x,y):X\times K\to\mathbb{R},\ E(x,y)=\frac{\sum_{g\in G}d(gf(g^{-1}x),y)^{2}}{|G|}.

For any fixed $x\in X$ , $E(x,\cdot)$ in strictly convex in $B_{\frac{1}{\epsilon}}(f(x))$ thus there is a global minimum point $y$ . Define $f_{G}(x)=y$ , then $f_{G}$ is $G$ -equivariant due to the uniqueness of the minimal point. $f_{G}$ is also a $\Phi(\epsilon|k,N)$ -GHA.

We next show that $f_{G}$ is $(k,\Phi(\epsilon|k,N))$ -splitting on every $\frac{1}{\epsilon}$ -ball by a similar argument in the proof of Theorem 3.3. For any $x_{0}\in X$ and $y_{0}=f_{G}(x_{0})$ . Define the energy function on the product space, $\tilde{E}:\prod_{l=1}^{|G|}B_{\frac{1}{\epsilon}}(y_{0})\times K\to\mathbb{R}$ by

\tilde{E}(\prod_{l=1}^{|G|}y_{l},y)=\frac{\sum_{g\in G}d(y_{l},y)^{2}}{|G|},y\in K,y_{l}\in B_{\frac{1}{\epsilon}}(y_{0}),l=1,...,|G|.

Then define $\tilde{h}(\prod_{l=1}^{|G|}y_{l})$ to be the center of mass with respect to $\tilde{E}$ . Then $f_{G}(x)=\tilde{h}(\prod_{g\in G}gf(g^{-1}x))$ by the definition.

Now consider the center of mass on the Euclidean space, which is a linear average. Define $\bar{h}:\prod_{l=1}^{|G|}B_{\frac{1}{\epsilon}}(y_{0})\to T_{y_{0}}K$ by $\bar{h}(\prod_{l=1}^{|G|}y_{l})=\frac{\sum_{l=1}^{|G|}\mathrm{exp}_{y_{0}}^{-1}(y_{l})}{|G|}$ . Then $\mathrm{exp}_{y_{0}}^{-1}\circ\tilde{h}$ is $\Phi(\epsilon|k,N)$ - $C^{3}$ -close to $\bar{h}$ as the metric on $B_{\frac{10}{\epsilon}}(y_{0})$ is $\epsilon^{2}$ - $C^{4}$ -close the a $B_{\frac{10}{\epsilon}}$ ball in $T_{y_{0}}K$ with the flat metric.

The linear average $\bar{h}(\prod_{g\in G}gf(g^{-1}x)):B_{\frac{1}{\epsilon}}(x_{0})\to T_{y_{0}}K$ is $(k,\Phi(\epsilon|k,N))$ -splitting by a similar argument as in Theorem 3.3, therefore

\mathrm{exp}_{y_{0}}^{-1}\circ f_{G}=\mathrm{exp}_{y_{0}}^{-1}\circ\tilde{h}(\prod_{g\in G}gf(g^{-1}x))

is also $(k,\Phi(\epsilon|k,N))$ -splitting on $B_{\frac{1}{\epsilon}}(x_{0})$ . ∎

4. Proof of Theorem A: construct an infranil-manifold

We prove Theorem A in this section.

Theorem A.

Given $N,v>0$ , there exists $\epsilon(N,v)>0$ and $C(N)>0$ , so that if a RCD $(-(N-1),N)$ space $(X,d,\mathcal{H}^{N})$ satisfies:

\mathrm{diam}(X)<\epsilon\leq\epsilon(N,v),\ \mathcal{H}^{N}(B_{1}(\tilde{p}))\geq v,\ \forall\tilde{p}\in\widetilde{X},

then $X$ is biHölder homeomorphic to an infranil-manifold $\mathcal{N}/\Gamma$ where $\mathcal{N}$ has a left-invariant metric, i.e., there exists $f:X\to\mathcal{N}/\Gamma$ with

(1-\Phi(\epsilon|N,v))d(x,y)^{1+\Phi(\epsilon|N,v)}\leq d(f(x),f(y))\leq(1+\Phi(\epsilon|N,v))d(x,y),

where $\Phi(\epsilon|N,v)\to 0$ as $\epsilon\to 0$ . Moreover, if $X$ is a smooth $N$ -manifold with $\mathrm{Ric}\geq-(N-1)$ , then $f$ is a diffeomorphism.

Assume that Theorem A does not hold. Then we can find a sequence of RCD $(-(N-1),N)$ space $(X_{i},d_{i},\mathcal{H}^{N})$ with non-collapsing universal covers and $\mathrm{diam}(X_{i})\to 0$ , while these spaces are not biHölder to any infranil-manifold.

Blow up the sequence slowly and pass to a subsequence if necessary, we may assume that the universal covers $\widetilde{X}_{i}$ of $X_{i}$ converge to $\mathbb{R}^{N}$ . By Theorem 2.4, for all sufficiently large $i$ , $G_{i}^{\prime}=\pi_{1}(X_{i},p_{i})$ contains a normal nilpotent subgroup $G_{i}$ of index $\leq C(N)$ . Let $X_{i}^{\prime}=\widetilde{X}_{i}/G_{i}$ .

$\begin{CD}(\widetilde{X}_{i},\tilde{p}_{i},G_{i},G_{i}^{\prime})@>{eGH}>{}>(\mathbb{R}^{N},\tilde{p},G,G^{\prime})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (X_{i}^{\prime},p_{i}^{\prime},G_{i}^{\prime}/G_{i})@>{GH}>{}>\mathrm{pt}\\ @V{}V{\pi}V@V{}V{\pi}V\\ (X_{i},p_{i})@>{GH}>{}>\mathrm{pt}\end{CD}$

$X_{i}^{\prime}$ converges to a point as it is a finite cover of $X_{i}$ with order $\leq C$ , thus $\mathrm{diam}(X_{i}^{\prime})\to 0$ .

Lemma 4.1.

$G_{i}$ contains no small subgroup, i.e., there exists $\delta>0$ such that the set

G_{i}(\delta)=\{g\in G_{i}^{\prime}|d(x,gx)<\delta,x\in B_{1/\delta}(\tilde{p}_{i})\}

contains no non-trivial subgroup for all large $i$ .

Proof.

Otherwise assume that $H_{i}$ is a non-trivial subgroup in $G_{i}(1/i)$ . Then $H_{i}$ converges to the identity, therefore $\widetilde{X}_{i}/H_{i}$ converges to $\mathbb{R}^{N}$ as well. By volume convergence theorem, the volume of $1$ -ball at $\pi(\tilde{p})$ in $\widetilde{X}_{i}/H_{i}$ is close to the volume of $1$ -ball at $\tilde{p}$ in $\widetilde{X}_{i}$ , a contradiction since $H_{i}$ is a small non-trivial subgroup. ∎

Lemma 4.2.

$G$ is free. In particular, $G$ can be identified as $\mathbb{R}^{N}$ .

Proof.

Consider any isotropy group

G_{x}=\{g\in G|gx=x\}

where $x\in\mathbb{R}^{N}$ . Then $G_{x}$ is compact. Since $G$ is a nilpotent Lie group, thus $[G_{x},G_{0}]$ is trivial, where $G_{0}$ is the identity component of $G$ . Since $G$ is transitive, then $G_{0}$ is also transitive. Therefore $G_{x}$ fixes all points, thus is a trivial group. ∎

$G$ has no isotropy. Thus if any $g\in G_{i}$ which moves $\tilde{p}_{i}$ small, $g$ must be close to the identity action.

Corollary 4.3.

For any $\delta>0$ , there exists $\epsilon>0$ such that $G_{i}(\tilde{p}_{i},\epsilon)\subset G_{i}(\delta)$ for sufficiently large $i$ .

Lemma 4.4.

For sufficiently large $i$ , $G_{i}$ is isomorphic to a lattice in a $N$ -dim simply connected nilpotent Lie group $\mathcal{N}_{i}$ .

Proof.

$G_{i}$ contains no small subgroup and converges to $\mathbb{R}^{N}$ . By Theorem 2.26, $G_{i}$ contains a nilprogession $P_{i}$ of dimension $N$ , which contains $G_{i}(\delta)$ for some $\delta>0$ . By Corollary 4.3, the nilprogession $P_{i}$ contains $G_{i}(\tilde{p}_{i},\epsilon)$ for some $\epsilon>0$ . We choose $i$ large enough such that diam $(X_{i})<\frac{\epsilon}{20}$ , then the groupfication of the $P_{i}$ is isomorphic to $G_{i}$ by Theorem 2.8. On the other hand, for $i$ large enough, thick $(P_{i})$ is large enough so that Theorem 2.23 holds, thus the groupfication of the $P_{i}$ is a lattice in a simply connected $N$ -dim nilpotent group $\mathcal{N}_{i}$ . Thus $G_{i}$ is isomorphic to a lattice in $\mathcal{N}_{i}$ . ∎

We shall use Theorem 2.30 to find a left-invariant metric on $\mathcal{N}_{i}$ so that it is locally $C^{4}$ -close to flat $\mathbb{R}^{N}$ .

Lemma 4.5.

For any $\epsilon\leq 1$ and $i$ large enough, $\mathcal{N}_{i}$ admits a left-invariant metric $g_{\mathcal{N}_{i}}$ with $\mathrm{inj}_{\mathcal{N}_{i}}\geq\frac{1}{\epsilon}$ . Moreover, there exists $\epsilon_{i}\to 0$ so that $\forall g\in\mathcal{N}_{i}$ , $B_{\frac{1}{\epsilon}}(g)\subset\mathcal{N}_{i}$ is $\epsilon_{i}$ - $C^{4}$ -close, by $\mathrm{exp}_{g}^{-1}$ , to the $\frac{1}{\epsilon}$ -ball in the tangent space $T_{g}\mathcal{N}_{i}$ with the flat metric.

Proof.

We always assume $\epsilon_{i}$ to be a sequence of numbers converging to $0$ while the value of $\epsilon_{i}$ depends on the specific setting. Consider the Lie algebra structure on $\mathcal{N}_{i}$ using the same notations in the argument after Theorem 2.26. Take $\{v_{i,j},1\leq j\leq N\}$ as a strong Malcev basis of the Lie algebra of $\mathcal{N}_{i}$ , then $\mathrm{exp}_{\mathcal{N}_{i}}(v_{i,j})=g_{i,j}\in\mathcal{N}_{i}\cap G_{i}$ . By our assumption, $g_{i,j}$ action on $\widetilde{X}_{i}$ pointed equivariantly converges to $g_{j}\in G=\mathbb{R}^{N}$ for any $1\leq j\leq N$ and $i\to\infty$ . $\{g_{j}\}_{j=1,2...,N}$ is a basis of $\mathbb{R}^{N}$ . We can take the corresponding Lie algebra $v_{j}=g_{j}$ since $G=\mathbb{R}^{N}$ is abelian. Define the left-invariant metric $g_{\mathcal{N}_{i}}$ by

g_{\mathcal{N}_{i}}(v_{i,j_{1}},v_{i,j_{2}})=\langle v_{j_{1}},v_{j_{2}}\rangle,

for any $1\leq j_{1},j_{2}\leq N$ and the right-hand side is the inner product in $\mathbb{R}^{N}$ .

Since $\{v_{i,j},1\leq j\leq n\}$ is a strong Malcev basis, for any $1\leq j_{1}<j_{2}\leq N$ ,

[v_{i,j_{1}},v_{i,j_{2}}]=\sum_{j=j_{2}+1}^{n}a_{i,j_{1}j_{2}}^{j}v_{i,j}.

$a_{i,j_{1}j_{2}}^{j}\to 0$ as $i\to 0$ by Theorem 2.30 and the fact that the limit group is abelian. Define $a_{i,j_{1}j_{2}}^{j}=0$ if $j\leq j_{1}$ or $j\leq j_{2}$ . Then by Koszul’s formula, for any $1\leq j_{1},j_{2},j_{3}\leq N$ ,

g_{\mathcal{N}_{i}}(\nabla_{v_{i,j_{1}}}v_{i,j_{2}},v_{i,j_{3}})=\frac{1}{2}(a_{i,j_{1}j_{2}}^{j_{3}}-a_{i,j_{2}j_{3}}^{j_{1}}+a_{i,j_{3}j_{1}}^{j_{2}}).

Since all terms on the right-hand side are constant (depending on $i$ ) and converge to $0$ as $i\to\infty$ . In particular, the covariant derivatives of the Riemannian curvature tensor $g_{\mathcal{N}_{i}}$ satisfy

|(\nabla^{g_{\mathcal{N}_{i}}})^{k}R_{g_{\mathcal{N}_{i}}}|\leq\epsilon_{i},\ 0\leq k\leq 3

where $\epsilon_{i}\to 0$ .

The sectional curvature of $\mathcal{N}_{i}$ is bounded by $\epsilon_{i}\to 0$ . By Theorem 2.29, $\mathcal{N}_{i}$ is diffeomorphiic to $\mathbb{R}^{N}$ . By our construction of the metric, $B_{\frac{4}{\epsilon}}(e)\subset\mathcal{N}_{i}$ must be biLipschitz on $B_{\frac{4}{\epsilon}}(0^{N})$ . Therefore the injective radius of $\mathcal{N}_{i}$ is at least $\frac{1}{\epsilon}$ for sufficiently large $i$ . The $C^{4}$ -closeness follows from the fact that $|(\nabla^{g_{\mathcal{N}_{i}}})^{k}R_{g_{\mathcal{N}_{i}}}|\leq\epsilon_{i}$ , $0\leq k\leq 3$ . ∎

From now on we always assume that $\mathcal{N}_{i}$ has the metric $g_{\mathcal{N}_{i}}$ constructed in Lemma 4.5.

Lemma 4.6 (Local eGH closeness).

For any $\epsilon>0$ , let $B_{\frac{1}{\epsilon}}(\tilde{p}_{i})\subset\widetilde{X}_{i}$ and $B_{\frac{1}{\epsilon}}(e)\subset\mathcal{N}_{i}$ . Then there exists an $\epsilon_{i}$ -GHA $h_{i}^{\prime}:B_{\frac{1}{\epsilon}}(\tilde{p}_{i})\to B_{\frac{1}{\epsilon}}(e)$ which is $\epsilon_{i}$ -almost $G_{i}(\tilde{p}_{i},{\frac{1}{\epsilon}})$ -equivariant if it is well-defined, where $\epsilon_{i}\to 0$ as $i\to 0$ .

Proof.

Take a linear map

\psi_{i}:\mathbb{R}^{N}\to\mathbb{R}^{N},\ \phi(x_{1},x_{2},...,x_{N})=\sum_{j=1}^{N}x_{j}v_{i,j}.

Take $\phi_{i}:\mathbb{R}^{N}\to\mathcal{N}_{i}$ as in Theorem 2.29 (2). By Theorem 2.30 and the definition of the metric in Lemma 4.5,

\phi_{i}\circ\psi_{i}^{-1}:\mathbb{R}^{N}\to\mathcal{N}_{i}

is a $\epsilon$ -GHA on $B_{\frac{1}{\epsilon}}(0^{N})$ for sufficiently large $i$ .

$(\widetilde{X}_{i},\tilde{p}_{i})$ is pGH-close to $\mathbb{R}^{N}$ by our assumption, and $\phi_{i}\circ\psi_{i}^{-1}:\mathbb{R}^{N}\to\mathcal{N}_{i}$ is a GHA on the $\frac{1}{\epsilon}$ -ball for all large $i$ . Combine two GHAs, we get $h_{i}^{\prime}:B_{\frac{1}{\epsilon}}(\tilde{p}_{i})\to B_{\frac{1}{\epsilon}}(e)$ which is a $\epsilon_{i}$ -GHA. We need to check $h_{i}^{\prime}$ is almost $G_{i}(\tilde{p}_{i},{\frac{1}{\epsilon}})$ -equivariant.

By our construction of $g_{i,j}$ (see the argument after Theorem 2.26), $\mathrm{exp}(v_{i,j})=g_{i,j}$ action on $\tilde{X}_{i}$ is $\epsilon_{i}$ -close to $g_{j}\in G=\mathbb{R}^{N}$ , $1\leq j\leq N$ . On the other hand, under the map $\phi_{i}\circ\psi_{i}^{-1}:\mathbb{R}^{N}\to\mathcal{N}_{i}$ , $g_{i,j}$ action on $\mathcal{N}_{i}$ is $\epsilon_{i}$ -close to the $g_{j}\in G=\mathbb{R}^{N}$ action. Thus $g_{i,j}$ actions on $B_{\frac{1}{\epsilon}}(\tilde{p}_{i})$ and $B_{\frac{1}{\epsilon}}(e)$ are $\epsilon_{i}$ -close to each other, that is, $h_{i}^{\prime}(g_{i,j}x)$ is $\epsilon_{i}$ -close to $g_{i,j}h_{i}^{\prime}(x)$ for all $x\in B_{\frac{1}{\epsilon}}(\tilde{p}_{i})$ such that $g_{i,j}x\in B_{\frac{1}{\epsilon}}(\tilde{p}_{i})$ , $1\leq j\leq N$ .

Take $C_{0}$ such that $|\psi_{i}^{-1}(B_{\frac{1}{\epsilon}}(0^{N}))|\leq\frac{C_{0}}{2}$ . Fix $j$ , recall $g_{i,j}=u_{i,j}^{l_{j}}$ where $u_{i,j}$ is one of the generators of the nilprogression and $l_{j}=\lfloor C_{j}/C\rfloor$ . Since the group actions $\langle u_{i,j}\rangle$ on both spaces $\widetilde{X}_{i}$ or $\mathcal{N}_{i}$ converge to a line in $G=\mathbb{R}^{N}$ . Then for any $k\leq C_{0}$ , $u_{i,j}^{k}$ action on $B_{\frac{1}{\epsilon}}(\tilde{p}_{i})$ and $B_{\frac{1}{\epsilon}}(e)$ are both $\epsilon_{i}$ -close to $g_{j}^{k/l_{j}}\in\mathbb{R}^{N}$ action.

For a general $g=u_{i,1}^{k_{1}}...u_{i,N}^{k_{N}}\in G_{i}(\tilde{p}_{i},\frac{1}{\epsilon})$ , we have $k_{j}\leq C_{0}$ , $1\leq j\leq N$ . Then any component of $g$ actions, saying $u_{i,1}^{k_{1}}$ , on $B_{\frac{1}{\epsilon}}(\tilde{p}_{i})$ and $B_{\frac{1}{\epsilon}}(e)$ are $\epsilon_{i}$ -close to each other. Therefore any $g\in G_{i}(\tilde{p}_{i},\frac{1}{\epsilon})$ actions on $B_{\frac{1}{\epsilon}}(\tilde{p}_{i})$ and $B_{\frac{1}{\epsilon}}(e)$ are $\epsilon_{i}$ -close to each other. ∎

By Lemma 4.6, we have a local GHA $h_{i}^{\prime}$ defined from $B_{\frac{1}{\epsilon}}(\tilde{p}_{i})$ to $B_{\frac{1}{\epsilon}}(e)\subset\mathcal{N}_{i}$ , which is almost $G_{i}(\tilde{p_{i}},\frac{1}{\epsilon})$ -equivariant. We can extend $h_{i}^{\prime}$ to a global map by the follows. For any $x\in\widetilde{X}_{i}$ , choose $g\in G_{i}$ so that $gx\in B_{1}(\tilde{p}_{i})$ . This choice is valid since $\widetilde{X}/G_{i}$ has a small diameter. Define $h_{i}$ by $h_{i}(x)=g^{-1}h^{\prime}(gx)\in\mathcal{N}_{i}$ . Note that different choices of $g$ yield only minor differences in $h_{i}(x)$ , so we can select one for our construction without loss of generality.

Lemma 4.7 (Extend a local approximation to the global map, [40]).

The map $h_{i}:\widetilde{X}_{i}\to\mathcal{N}_{i}$ is a global map which is an $\epsilon_{i}$ -GHA on any $\frac{1}{\epsilon}$ -ball and $\epsilon_{i}$ -almost $G_{i}$ -equivariant, for some $\epsilon_{i}\to 0$ .

It is well known that a subgroup of finite index contains a normal subgroup of finite index. We provide a proof for readers’ convenience.

Lemma 4.8.

Assume that group $H$ contains a subgroup $H_{0}$ of index $|H:H_{0}|\leq C$ , then there is normal subgroup $H^{\prime}$ of $H$ such that $|H/H^{\prime}|\leq C!$ and $H^{\prime}\subset H_{0}$ .

Proof.

Define $H/H_{0}$ be the set of left cosets of $H_{0}$ in $H$ , the index is $\leq C$ . Define a homomorphism $\phi:H\to\mathrm{sym}(H/H_{0})$ by $\phi(g,g^{\prime}H_{0})=(gg^{\prime})H_{0}$ . Then define $H^{\prime}=\mathrm{Ker}(\phi)$ , which is a normal subgroup of $H$ with index less than or equal to the order of the symmetric group $\mathrm{sym}(H/H_{0})$ , thus $|H/H^{\prime}|\leq C!$ .

For any $g\in H^{\prime}$ , since $\phi(g,H_{0})=H_{0}$ , it follows that $g\in H_{0}$ . Then $H^{\prime}\subset H_{0}$ . ∎

Now we can complete the proof of Theorem A.

Proof of Theorem A.

Consider a contradiction sequence,

We assumed that none of $X_{i}$ is biHölder to an infranil-manifold.

We have established that $G_{i}$ acts as lattice in a simply connected nilpotent Lie group $\mathcal{N}_{i}$ . Since $G_{i}$ is a normal subgroup of $G_{i}^{\prime}$ with bounded index, by the rigidity result from [30], $G_{i}^{\prime}$ can be identified as a discrete subset of $\mathcal{N}_{i}\rtimes\text{Aut}(\mathcal{N}_{i})$ . Since the lattice $G_{i}$ is $\epsilon_{i}$ -dense in $\mathcal{N}_{i}$ , $h_{i}$ in Lemma 4.7 is also $\epsilon_{i}$ -almost $G_{i}^{\prime}$ -equivariant by the rigidity.

Since $g_{\mathcal{N}_{i}}$ is left-invariant metric on $\mathcal{N}_{i}$ , $G_{i}$ actions on $\mathcal{N}_{i}$ are isometric. For any $g^{\prime}=(g,\phi)=G_{i}^{\prime}\subset\mathcal{N}_{i}\rtimes\text{Aut}(\mathcal{N}_{i})$ where $g\in\mathcal{N}_{i}$ and $\phi\in\text{Aut}(\mathcal{N}_{i})$ , $g^{\prime}$ action on either $(\widetilde{X}_{i},\tilde{p}_{i})$ or $(\mathcal{N}_{i},e)$ is eGH close to an isometric action on $\mathbb{R}^{N}$ . Since $(\mathcal{N}_{i},e)$ pointed $C^{5}$ -close to flat $\mathbb{R}^{N}$ , $\phi^{*}(g_{\mathcal{N}_{i}})$ must be $C^{4}$ close to $g_{\mathcal{N}_{i}}$ . We have $|G_{i}^{\prime}/G_{i}|\leq C$ , thus the number of the choices of $\phi$ is at most $C$ . Averaging all such $\phi^{*}(g_{\mathcal{N}_{i}})$ if necessary, we may simply assume that $\phi^{*}(g_{\mathcal{N}_{i}})=g_{\mathcal{N}_{i}}$ , then $G_{i}^{\prime}$ actions on $\mathcal{N}_{i}$ are isometric.

Take any small $\epsilon>0$ . The finitely generated nilpotent group $G_{i}$ is residually finite, i.e., for any $g\in G_{i}$ , there exists a finite index normal subgroup of $G_{i}$ which does not contain $g$ . Therefore there is a normal subgroup $G_{i}^{\prime\prime}$ of $G_{i}$ with finite index so that $G_{i}^{\prime\prime}\cap B_{\frac{1}{\epsilon}}(e)=\emptyset$ . We may also assume that $G_{i}^{\prime\prime}$ is normal in $G_{i}^{\prime}$ as well due to Lemma 4.8.

Since $G_{i}^{\prime\prime}\cap B_{\frac{1}{\epsilon}}(e)=\emptyset$ , we can apply Lemma 4.5 to conclude that the injective radius of $\mathcal{N}_{i}/G_{i}^{\prime\prime}$ is at least ${\frac{1}{\epsilon}}$ . For any $y\in\mathcal{N}_{i}/G_{i}^{\prime\prime}$ , $B_{\frac{1}{\epsilon}}(y)\subset\mathcal{N}_{i}/G_{i}^{\prime\prime}$ is $\epsilon_{i}$ - $C^{4}$ -close to the $\frac{1}{\epsilon}$ -ball in the tangent space $T_{y}(\mathcal{N}_{i}/G_{i}^{\prime\prime})$ with the flat metric.

Since $h_{i}$ is $\epsilon_{i}$ -almost $G_{i}^{\prime}$ -equivariant, we can reduce $h_{i}$ to a map

\bar{h}_{i}:\widetilde{X}_{i}/G_{i}^{\prime\prime}\to\mathcal{N}_{i}/G_{i}^{\prime\prime},

which is an $\epsilon_{i}$ -GHA on any ${\frac{1}{\epsilon}}$ -ball and $\epsilon_{i}$ -almost $G_{i}^{\prime}/G_{i}^{\prime\prime}$ -equivariant. For any $x\in\widetilde{X}_{i}/G_{i}^{\prime\prime}$ , define $\bar{h}_{i}(x)=\pi(h_{i}(\tilde{x}))\in\mathcal{N}_{i}/G_{i}^{\prime\prime}$ , where $\tilde{x}$ is a pre-image of $x$ in $\widetilde{X}_{i}$ . Different choices of $\tilde{x}$ lead to only minor differences in $\bar{h}_{i}$ since $h_{i}$ is almost $G_{i}^{\prime}$ -equivariant.

Since $G_{i}^{\prime}/G_{i}^{\prime\prime}$ is finite, we can apply Theorem 3.5 to

\bar{h}_{i}:(\widetilde{X}_{i}/G_{i}^{\prime\prime},G_{i}^{\prime}/G_{i}^{\prime\prime})\longrightarrow(\mathcal{N}_{i}/G_{i}^{\prime\prime},G_{i}^{\prime}/G_{i}^{\prime\prime}).

We can find a $(G_{i}^{\prime}/G_{i}^{\prime\prime})$ -equivariant map $f_{G_{i}^{\prime}/G_{i}^{\prime\prime}}$ from $\widetilde{X}_{i}/G_{i}^{\prime\prime}$ to $\mathcal{N}_{i}/G_{i}^{\prime\prime}$ which is $(N,\Phi(\epsilon|N))$ -splitting on any $\frac{1}{10\epsilon}$ -ball. Then we have biHölder estimates from [22],

(1-\Phi(\epsilon|N))d_{i}(x,y)^{1+\Phi(\epsilon|N)}\leq d(f_{G_{i}^{\prime}/G_{i}^{\prime\prime}}(x),f_{G_{i}^{\prime}/G_{i}^{\prime\prime}}(y))\leq(1+\Phi(\epsilon|N))d_{i}(x,y),

for any $x,y\in\widetilde{X}_{i}/G_{i}^{\prime\prime}$ with $d_{i}(x,y)\leq\frac{1}{20\epsilon}$ .

Since $f_{G_{i}^{\prime}/G_{i}^{\prime\prime}}$ is $(G_{i}^{\prime}/G_{i}^{\prime\prime})$ -equivariant, it can be naturally reduced to a biHölder map on the quotient space $f:X_{i}=\widetilde{X}_{i}/G_{i}^{\prime}\to\mathcal{N}_{i}/G_{i}^{\prime}$ . Thus $X_{i}$ is biHölder to an infranil-manifold, a contradiction to the assumption. Moreover, if $X_{i}$ is a smooth manifold with $\mathrm{Ric}\geq-(N-1)$ , then $f$ is smooth and $df$ is non-degenerate, thus $X_{i}$ is biHölder diffeomorphic to $\mathcal{N}_{i}/G_{i}^{\prime}$ . ∎

Remark 4.9.

At the beginning of the proof of Theorem A, we slowly blow up the metric to get a limit space $\mathbb{R}^{N}$ . Therefore, for a contradiction sequence $(X_{i},d_{i},\mathcal{H}^{N})$ , we actually proved the biHölder estimate for $(X_{i},r_{i}d_{i},\mathcal{H}^{N})$ , where $r_{i}\to\infty$ slowly,

(1-\Phi(\epsilon|N))(r_{i}d_{i}(x,y))^{1+\Phi(\epsilon|N)}\leq r_{i}d(f(x),f(y))\leq(1+\Phi(\epsilon|N))r_{i}d_{i}(x,y)

where $d$ is the distance function on $\mathcal{N}_{i}$ . Fix a large $i$ , we can take $d^{\prime}=\frac{d}{r_{i}}$ on $\mathcal{N}_{i}$ . Since

(1-\Phi(\epsilon|N))(r_{i}d_{i}(x,y))^{1+\Phi(\epsilon|N)}\geq r_{i}(1-\Phi(\epsilon|N))(d_{i}(x,y))^{1+\Phi(\epsilon|N)},

we have

(1-\Phi(\epsilon|N))(d_{i}(x,y))^{1+\Phi(\epsilon|N)}\leq d^{\prime}(f(x),f(y))\leq(1+\Phi(\epsilon|N))d_{i}(x,y).

Then $f:(X_{i},d_{i})\to(\mathcal{N}_{i},d^{\prime})$ is also biHölder. Thus the biHölder estimate on the blowing up metric implies the biHölder estimate on the original metric. For this reason, in the next sections we shall omit some blowing up arguments and directly apply Theorem 3.5 and canonical Reifenberg method; see the proofs of Lemma 6.3 and Theorem B.

5. Mixed curvature and almost flat manifolds theorem

In this section we prove Theorem 1.7 using a similar construction in the proof of Theorem A. The main difference is that we need to glue strainer maps instead of almost splitting maps.

Take small $\epsilon>0$ . Assume that $(X,d,\mathfrak{m},p)$ is an $\mathrm{RCD}(-\epsilon,N)$ space such that $(X,d)$ is $\mathrm{CBA}(\epsilon)$ , $\partial X=\emptyset$ . Suppose that $(K,q)$ is a smooth Riemannian $n$ -manifold with $\mathrm{inj}_{K}\geq\frac{10}{\epsilon}$ . $B_{\frac{10}{\epsilon}}(p)$ is $\epsilon^{2}$ - $C^{4}$ -close, by $\mathrm{exp}_{p}^{-1}$ , to its preimage in the tangent space $T_{p}K$ with the flat metric. Suppose that $h:B_{2}(p)\to B_{2}(q)$ is an $\epsilon$ -GHA.

By [26], we can use a strainer to construct a differentiable $\Phi(\epsilon|n)$ -GHA $u:B_{1}(p)\to B_{2}(q)$ with $1-\Phi(\epsilon|n)\leq|\nabla u|\leq 1+\Phi(\epsilon|n)$ . In particular, $u$ is diffeomorphic onto its image. Moreover, if we use another strainer to construct another differentiable $\Phi(\epsilon|n)$ -GHA, saying $u^{\prime}:B_{1}(p)\to B_{2}(q)$ . Then $u^{\prime}$ is $\Phi(\epsilon|n)$ - $C^{1}$ -close to $u$ .

By the same construction in Theorem 3.5, we can glue local strainer maps and obtain the following theorem; see also the proof of the fibration theorem for mixed curvature spaces in [26].

Theorem 5.1.

Given $N$ and $n\leq N$ , there exists $\epsilon(N,n)$ such that for any $\epsilon\leq\epsilon(N,n)$ , the following holds. Assume that a smooth $n$ -manifold $K$ satisfies $\mathrm{inj}_{K}\geq\frac{10}{\epsilon}$ and for any $p\in K$ , $B_{\frac{10}{\epsilon}}(p)$ is $\epsilon^{2}$ - $C^{4}$ -close, by $\mathrm{exp}_{p}^{-1}$ , to its preimage in the tangent space $T_{p}K$ with the flat metric. Suppose that $(X,d,\mathfrak{m})$ is an $\mathrm{RCD}(-\epsilon,N)$ space such that $(X,d)$ is $\mathrm{CBA}(\epsilon)$ , $\partial X=\emptyset$ , $\mathrm{dim}(X)=n$ . Assume that there is a global map $h:X\to K$ which is an $\epsilon$ -GHA map on any $\frac{10}{\epsilon}$ -ball in $X$ ; a finite group $G$ acts isometrically on $X$ and $K$ separately and $h$ is $\epsilon$ -almost $G$ -equivariant.

Then there is a $G$ -equivariant map $f_{G}:X\to K$ , which is also a $\Phi(\epsilon|n,N)$ -GHA and biLipschitz diffeomorphic on any $\frac{1}{\epsilon}$ -ball, that is, for any $x,y\in X$ with $d(x,y)<\frac{1}{\epsilon}$ ,

(1-\Phi(\epsilon|n,N))d(x,y)\leq d(f_{G}(x),f_{G}(y))\leq(1+\Phi(\epsilon|n,N))d(x,y),

where $\Phi(\epsilon|n,N)\to 0$ as $\epsilon\to 0$ .

We now proceed to prove Theorem 1.7 using Theorem 5.1 and the construction in the proof of Theorem A.

Proof of Theorem 1.7.

We apply a contradiction argument. Assume that there exists $\epsilon_{i}\to 0$ so that there exists $(X_{i},d_{i},\mathfrak{m}_{i})$ which is $\mathrm{RCD}(-\epsilon_{i},N)$ and $\mathrm{CBA}(\epsilon_{i})$ ; $\partial X_{i}=\emptyset$ and $\mathrm{diam}(X_{i})\leq\epsilon_{i}$ . And we assume that none of $X_{i}$ is biLipschitz diffeomorphic to an infranil-manifold of dimension $\leq N$ . We need to prove that $X_{i}$ is biLipschitz diffeomorphic to an infranil-manifold of dimension $\leq N$ for some large $i$ .

Passing to a subsequence if necessary, we may assume that all $X_{i}$ are manifolds of dimension $n$ where $n\leq N$ . We recall the argument from [26] that the universal cover $\widetilde{X}_{i}$ of $X_{i}$ pGH converges to $\mathbb{R}^{n}$ . Specifically, take $p_{i}\in X_{i}$ and fix a large $r>0$ . We consider $\hat{X}_{i}=B_{r}(0^{n})\subset T_{p_{i}}X_{i}$ , the $r$ -ball centered at the origin in the tangent space $T_{p_{i}}X_{i}$ with the pull back metric. Since the curvature of $X_{i}$ is bounded above by $\epsilon_{i}\to 0$ , the exponential map $\mathrm{exp}_{p_{i}}:\hat{X}_{i}\to X_{i}$ is non-degenerate for sufficiently large $i$ . Thus $\hat{X}_{i}$ a pseudo-cover of $X_{i}$ with pseudo-actions. By Lemma 3.7 in [26] $\hat{X}_{i}$ pGH converges to a $r$ -ball in $\mathbb{R}^{n}$ .

Then by Lemma 2.5 in [26], the groupfication of pseudo-actions is exactly $\pi_{1}(X_{i})$ . Therefore the gluing space using the pseudo-cover and pseudo-actions (see section 2.3) is exactly the universal cover $\widetilde{X}_{i}$ which also pGH converges to $\mathbb{R}^{n}$ .

Let $\tilde{p}_{i}$ be a lift of $p_{i}$ in $\widetilde{X}_{i}$ . By the generalized Margulis lemma, for sufficiently large $i$ , $G_{i}^{\prime}=\pi_{1}(X_{i},p_{i})$ contains a nilpotent subgroup $G_{i}$ with index $\leq C$ . Properly rescale the measure on $\widetilde{X}_{i}$ and let $X_{i}^{\prime}=\widetilde{X}_{i}/G_{i}$ . By the splitting theorem, we have the following diagram.

$\begin{CD}(\widetilde{X}_{i},\tilde{p}_{i},G_{i},G_{i}^{\prime},\mathfrak{m}_{i})@>{epmGH}>{}>(\mathbb{R}^{n},\tilde{p},G,G^{\prime},\mathcal{H}^{n})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (X_{i}^{\prime},p_{i}^{\prime},G_{i}^{\prime}/G_{i})@>{GH}>{}>\mathrm{pt}\\ @V{}V{\pi}V@V{}V{\pi}V\\ (X_{i},p_{i})@>{GH}>{}>\mathrm{pt}\end{CD}$

By Lemma 4.2, $G$ is free, thus can be identified as $\mathbb{R}^{n}$ .

$G_{i}$ has no small subgroup due to the measure convergence. Then by Lemma 4.4, for sufficiently large $i$ , $G_{i}$ is isomorphic to a lattice in a $n$ -dim simply connected nilpotent Lie group $\mathcal{N}_{i}$ . For any $\epsilon>0$ and sufficiently large $i$ , by Lemma 4.5, we can endow $\mathcal{N}_{i}$ a left-invariant metric so that $\mathcal{N}_{i}$ is $\epsilon$ - $C^{4}$ -close to $\mathbb{R}^{n}$ . Next we apply Lemma 4.6 and 4.7, we can find a map $h_{i}:\widetilde{X}_{i}\to\mathcal{N}_{i}$ so that , $h_{i}$ is an $\epsilon_{i}$ -GHA on any $\frac{1}{\epsilon}$ -ball in $\widetilde{X}_{i}$ and $\epsilon_{i}$ -almost $G_{i}$ -equivariant. By the rigidity, $G_{i}^{\prime}$ can be identified as a discrete subset of $\mathcal{N}_{i}\rtimes\text{Aut}(\mathcal{N}_{i})$ . Then $h_{i}$ is also $\epsilon_{i}$ -almost $G_{i}^{\prime}$ -equivariant.

We can find a normal subgroup $G_{i}^{\prime\prime}$ of $G_{i}^{\prime}$ with finite index so that $G_{i}^{\prime\prime}\subset G_{i}$ and $G_{i}^{\prime\prime}\cap B_{\frac{1}{\epsilon}}(e)=\emptyset$ where $e\in\mathcal{N}_{i}$ . In particular, $\mathcal{N}_{i}/G_{i}^{\prime\prime}$ is compact and $\epsilon$ - $C^{4}$ -close to $\mathbb{R}^{n}$ on any $\frac{1}{\epsilon}$ -ball. Then we can find a map $\bar{h}_{i}:\widetilde{X}_{i}/G_{i}^{\prime\prime}\to\mathcal{N}_{i}/G_{i}^{\prime\prime}$ , which is $\epsilon_{i}$ -GHA on any $\frac{1}{\epsilon}$ -ball and $\epsilon_{i}$ -almost $G_{i}^{\prime}/G_{i}^{\prime\prime}$ -equivariant. Then applying Theorem 5.1 to $\bar{h}_{i}$ , we can obtain an $G_{i}^{\prime}/G_{i}^{\prime\prime}$ -equivariant GHA map $f_{G_{i}^{\prime}/G_{i}^{\prime\prime}}:\widetilde{X}_{i}/G_{i}^{\prime\prime}\to\mathcal{N}_{i}/G_{i}^{\prime\prime}$ , which is biLipschitz diffeomorphic on any $\frac{1}{\epsilon}$ -ball.

Since $f_{G_{i}^{\prime}/G_{i}^{\prime\prime}}$ is $G_{i}^{\prime}/G_{i}^{\prime\prime}$ -equivariant and the diameter of $X_{i}$ converges to $0$ , $f_{G_{i}^{\prime}/G_{i}^{\prime\prime}}$ induces a biLipschitz diffeomorphic map on the quotient space

f_{i}:X_{i}=\widetilde{X}_{i}/G_{i}^{\prime}\to\mathcal{N}_{i}/G_{i}^{\prime}.

Therefore $X_{i}$ is biLipschitz diffeomorphic to an infranil-manifold $\mathcal{N}_{i}/G_{i}^{\prime}$ . ∎

6. Proof of Theorem B: construct fibers along the collapsing direction

We restate Theorem B.

Theorem B.

Given $N,v>0$ , assume that a sequence of compact RCD $(-(N-1),N)$ spaces $(X_{i},d_{i},\mathcal{H}^{N})$ with $(1,v)$ -bound covering geometry converges to a $k$ -manifold $K$ in the Gromov-Hausdorff sense. Then for all large $i$ , there is a fiber bundle map $f_{i}:X_{i}\to K$ which is a GHA. Moreover, the fiber is homeomorphic to an infranil-manifold and the structure group is affine.

6.1. Basic constructions

To prove Theorem B, we first employ the construction from Theorem 3.3 to obtain an $\epsilon_{i}$ -GHA $f_{i}:X_{i}\to K$ which is locally almost $k$ -splitting. We want to prove that $f_{i}$ is actually a fibration map for sufficiently large $i$ . Assume that for some $p_{i}\in X_{i}$ converging to $p\in K$ , $f_{i}$ is not a fibration map on any small neighborhood of $p_{i}$ . We will show the existence of fiberation which leads to a contradiction.

By the generalized Margulis lemma, we can find $\epsilon_{0}>0$ such that $G_{i}^{\prime}$ , the image of $\pi_{1}(B_{\epsilon_{0}}(p_{i}))\to\pi_{1}(B_{1}(p_{i}))$ , contains a nilpotent subgroup $G_{i}$ of finite index $\leq C$ . Let $\widetilde{B}(p_{i},\epsilon_{0},1)$ be a connected component of the pre-image of $B_{\epsilon_{0}}(p_{i})$ in the universal cover of $B_{1}(p_{i})$ . Passing to a subsequence if necessary,

(\widetilde{B}(p_{i},\epsilon_{0},1),\tilde{p}_{i})\overset{pGH}{\longrightarrow}(Y,\tilde{p})

by the precomactness Theorem 2.9. Moreover, by [23], the tangent cone at $\tilde{x}\in Y$ must be $\mathbb{R}^{N}$ since any tangent cone at $K$ is $\mathbb{R}^{k}$ .

We use the notation $r_{i}B_{\epsilon_{0}}(p_{i})$ for the set $B_{\epsilon_{0}}(p_{i})\subset X_{i}$ with the rescaled metric $r_{i}d_{i}$ . Take $r_{i}\to\infty$ slowly to blow up the metric and pass to a subsequence if necessary,

(6.1)

\displaystyle\begin{CD}(r_{i}\widetilde{B}(p_{i},\epsilon_{0},1),\tilde{p}_{i},G_{i},G_{i}^{\prime})@>{eGH}>{}>(\mathbb{R}^{N},0^{N},G,G^{\prime})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/G_{i},p_{i}^{\prime},G_{i}^{\prime}/G_{i})@>{GH}>{}>(Y^{\prime},p^{\prime},\bar{G})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (r_{i}B_{\epsilon_{0}}(p_{i}),p_{i})@>{F_{i}}>{}>(\mathbb{R}^{k},0^{k})\end{CD}

where $F_{i}=\exp_{p}^{-1}\circ f_{i}$ is almost $k$ -splitting by Theorem 3.3. Since $G_{i}$ is a normal subgroup of $G_{i}^{\prime}$ with index $\leq C$ , $\widetilde{B}(p_{i},\epsilon_{0},1)/G_{i}$ is a finite cover of $B_{\epsilon_{0}}(p_{i})$ .

By the covering lemma 2.12,

\tilde{F}_{i}:r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)\overset{\pi}{\longrightarrow}r_{i}B_{\epsilon_{i}}(p_{i})\overset{F_{i}}{\longrightarrow}(\mathbb{R}^{k},0^{k}),

\tilde{F}_{i}^{\prime}:r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/G_{i}\overset{\pi}{\longrightarrow}r_{i}B_{\epsilon_{i}}(p_{i})\overset{F_{i}}{\longrightarrow}(\mathbb{R}^{k},0^{k})

are also almost $k$ -splitting. In particular, $Y^{\prime}=\mathbb{R}^{k}\times Y^{\prime\prime}$ and $\mathbb{R}^{N}=\mathbb{R}^{k}\times\mathbb{R}^{N-k}$ .

Thus $\bar{G},G,G^{\prime}$ actions are trivial on the first $\mathbb{R}^{k}$ component of $\mathbb{Y}^{\prime}$ and $\mathbb{R}^{N}$ respectively. In particular, since the order of $|G_{i}^{\prime}/G_{i}|\leq C$ , $Y^{\prime\prime}$ is a finite set. $Y^{\prime\prime}$ is connected, thus $Y^{\prime\prime}$ is a single point and $Y^{\prime}=\mathbb{R}^{k}$ . Then $\bar{G}$ is trivial; $G$ and $G^{\prime}$ act transitively on the $\mathbb{R}^{N-k}$ component of $\mathbb{R}^{N}=\mathbb{R}^{k}\times\mathbb{R}^{N-k}$ . Thus by the same proof of Lemma 4.2, we conclude that:

Lemma 6.1.

$G$ is free. In particular, $G$ can be identified as $\mathbb{R}^{N-k}$ .

By Lemma 4.3, $G_{i}$ contains no small subgroup. Now we can simplify diagram 6.1 as:

(6.2)

\displaystyle\begin{CD}(r_{i}\widetilde{B}(p_{i},\epsilon_{0},1),\tilde{p}_{i},G_{i},G_{i}^{\prime})@>{eGH}>{}>(\mathbb{R}^{N},0^{N},G=\mathbb{R}^{N-k},G^{\prime})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/G_{i},p_{i}^{\prime},G_{i}^{\prime}/G_{i})@>{eGH}>{}>(\mathbb{R}^{k},0^{k},\mathrm{id})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (r_{i}B_{\epsilon_{0}}(p_{i}),p_{i})@>{F_{i}}>{}>(\mathbb{R}^{k},0^{k})\end{CD}

The strategy for proving Theorem B follows a similar approach to the proof of Theorem A. Specifically, we use a nilpotent subgroup of the (local relative) fundamental group to construct a simply connected nilpotent Lie group. Then we show that the (local relative) covering space locally admits a fibration structure. However, in the case of Theorem B, there are two additional challenges compared to the proof of Theorem A.

The first challenge is that the generators of $G_{i}$ may be large in the rescaled metric spaces $r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)$ . The second challenge is that we cannot directly apply Theorem 2.8 to local relative fundamental groups.

From now on we always consider rescaled metrics $r_{i}d_{i}$ . We solve the first issue using the gap lemma and choosing a subgroup of $G_{i}$ .

For any $r>0$ , define

G_{i}(\tilde{p}_{i},r)=\{g\in G_{i}|r_{i}d_{i}(g\tilde{p}_{i},\hat{p}_{i})\leq r\}.

By the gap lemma 2.13 and Lemma 6.1, there exists $\epsilon_{i}\to 0$ with $\epsilon_{i}r_{i}\to\infty$ and for all $r\in[\epsilon_{i},\frac{1}{\epsilon_{i}}]$ , $\langle G_{i}(\hat{p}_{i},r)\rangle$ is the same group, saying $H_{i}$ . Then the equivariant limit of $H_{i}$ contains a neighborhood of $G=\mathbb{R}^{N-k}$ thus must equal $G$ itself.

We next prove that

G_{i}^{\prime}(\tilde{p}_{i},r)=\{g^{\prime}\in G_{i}^{\prime}|r_{i}d_{i}(g^{\prime}\tilde{p}_{i},\hat{p}_{i})\leq r\}

generates the same group in $G_{i}^{\prime}$ for all $r\in[2\epsilon_{i},\frac{1}{\epsilon_{i}}-\epsilon_{i}]$ as well, saying $H_{i}^{\prime}$ . For any $g^{\prime}\in G_{i}^{\prime}(\tilde{p},\frac{1}{\epsilon_{i}}-\epsilon_{i})$ , since $G_{i}/G_{i}^{\prime}$ converges to the trivial group, we can find $g\in G_{i}(\tilde{p}_{i},\frac{1}{\epsilon_{i}})$ so that $r_{i}d_{i}(g^{\prime}g^{-1}\tilde{p},\tilde{p})\leq 2\epsilon_{i}$ . Thus $g^{\prime}g^{-1}\in G_{i}^{\prime}(\tilde{p},2\epsilon_{i})$ . On the other hand $g\in H_{i}\subset\langle G_{i}^{\prime}(\tilde{p}_{i},2\epsilon_{i})\rangle$ thus $g^{\prime}=g^{\prime}g^{-1}g\in\langle G_{i}^{\prime}(\tilde{p}_{i},2\epsilon_{i})\rangle$ .

Lemma 6.2.

For sufficiently large $i$ , $H_{i}$ is a normal subgroup of $H_{i}^{\prime}$ with index $\leq C$ , where $C$ is the constant in the generalized Margulis lemma.

Proof.

$H_{i}^{\prime}=\langle G_{i}^{\prime}(\tilde{p}_{i},2\epsilon_{i})\rangle$ and $H_{i}=\langle G_{i}(\tilde{p}_{i},\epsilon_{i})\rangle$ . To show that $H_{i}$ is normal in $H_{i}^{\prime}$ , we only need to show that $h^{-1}gh\in H_{i}$ for any $h\in G_{i}^{\prime}(\tilde{p}_{i},2\epsilon_{i})$ and $g\in G_{i}(\tilde{p}_{i},\epsilon_{i})$ . Since $G_{i}$ is normal in $G_{i}^{\prime}$ , $h^{-1}gh\in G_{i}$ . Then

	$\displaystyle r_{i}d_{i}(\tilde{p}_{i},h^{-1}gh\tilde{p}_{i})$	$\displaystyle=r_{i}d_{i}(h\tilde{p}_{i},gh\tilde{p}_{i})$
		$\displaystyle\leq r_{i}d_{i}(h\tilde{p}_{i},\tilde{p}_{i})+r_{i}d_{i}(\tilde{p}_{i},g\tilde{p}_{i})+r_{i}d_{i}(g\tilde{p}_{i},gh\tilde{p}_{i})$
		$\displaystyle\leq 5\epsilon_{i}.$

Since $G_{i}(\tilde{p}_{i},5\epsilon_{i})\subset H_{i}$ , $h^{-1}gh\in H_{i}$ . Thus $H_{i}$ is a normal subgroup of $H_{i}^{\prime}$ .

Next we prove that $|H_{i}^{\prime}/H_{i}|\leq C$ . Assume $|H_{i}^{\prime}/H_{i}|>C\geq|G_{i}^{\prime}/G_{i}|$ . Let $T$ denote the image of $H_{i}^{\prime}(\tilde{p}_{i},2\epsilon_{i})$ in $H_{i}^{\prime}/H_{i}$ . Define $T^{2}=\{g_{1}g_{2}|g_{1},g_{2}\in T\}$ and similarly $T^{l}$ for $l>0$ . If $T^{l+1}=T^{l}$ for some $l>0$ , then $T^{l}=H_{i}^{\prime}/H_{i}$ . Therefore either $|T^{l+1}|\geq|T^{l}|+1$ or $|T^{l}|>C$ . In either case, we always have the order $|T^{C+1}|\geq C+1$ . However, since $\epsilon_{i}\to 0$ , $(H_{i}^{\prime}(\tilde{p}_{i},2\epsilon_{i}))^{C+1}\subset H_{i}^{\prime}(\tilde{p}_{i},1)$ for sufficiently large $i$ . Then we can find $g_{1},g_{2}\in H_{i}^{\prime}(\tilde{p}_{i},1)$ so that they have different images in $H_{i}^{\prime}/H_{i}$ but the same image in $G_{i}^{\prime}/G_{i}$ . Then $g_{1}g_{2}^{-1}\in G_{i}(\tilde{p}_{i},3)\subset H_{i}$ , a contradiction. ∎

Since any $g_{i}\in G_{i}$ but $g_{i}\notin H_{i}$ satisfies $r_{i}d_{i}(\tilde{p},g\tilde{p})>\frac{1}{\epsilon_{i}}$ , it diverges in the pointed Gromov-Hausdorff sense. Thus a large ball in $r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/G_{i}$ is isometric to a large ball in $r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/H_{i}$ . To simplify the notation, we can identify:

r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/G_{i}=r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/H_{i}.

Similar we can identify:

r_{i}B_{\epsilon_{0}}(p_{i})=r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/G_{i}^{\prime}=r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/H_{i}^{\prime}.

From 6.2, we have the following commutative diagram:

(6.3)

\displaystyle\begin{CD}(r_{i}\widetilde{B}(p_{i},\epsilon_{0},1),\tilde{p}_{i},H_{i},H_{i}^{\prime})@>{eGH}>{}>(\mathbb{R}^{N},0^{N},H=\mathbb{R}^{N-k},H^{\prime})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/H_{i},p_{i}^{\prime},H_{i}^{\prime}/H_{i})@>{eGH}>{}>(\mathbb{R}^{k},0^{k},\mathrm{id})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (r_{i}B_{\epsilon_{0}}(p_{i}),p_{i})@>{F_{i}}>{}>(\mathbb{R}^{k},0^{k})\end{CD}

6.2. The existence of local almost product structure

In this subsection, we always consider the rescaled metric $r_{i}d_{i}$ on $(X_{i},p_{i})$ . Let $B^{r_{i}}_{r}(p_{i})$ denote the $r$ -ball of $p_{i}$ with respect to the rescaled metric $r_{i}d_{i}$ . We will always use the rescaled metric $r_{i}d_{i}$ on $B^{r_{i}}_{r}(p_{i})$ .

The goal of this subsection is to prove the following result.

Lemma 6.3 (Existence of local product structure).

In the context of 6.3, there exists $r>0$ so that for sufficiently large $i$ , $B_{r}^{r_{i}}(p_{i}^{\prime})$ is, removing some points near the boundary if necessary, biHölder homeomorphic to $B_{r}(0^{k})\times\mathcal{N}_{i}/\hat{H}_{i}$ , where $B_{r}(0^{k})\subset\mathbb{R}^{k}$ and $\mathcal{N}_{i}$ is a simply connected nilpotent Lie group with lattice $\hat{H}_{i}$ .

Remark 6.4.

Lemma 6.3 claims the existence of a nil-manifold fibration structure on a neighborhood of $p_{i}^{\prime}$ (not $p_{i}$ ). The obstruction to constructing an infranil-manifold fibration structure near $p_{i}$ is that $H_{i}$ is not necessarily isomorphic to a lattice. This issue will be addressed in Lemma 6.7.

The proof of Lemma 6.3 follows by applying strategy used in the proof of Theorem A to the collapsing direction. In Theorem A, we first use Theorem 2.26 to find the nilprogression structure of a neighborhood of the identity in the fundamental group; then we apply Theorem 2.8 to show that the fundamental group is isomorphic to a lattice in a nilpotent Lie group. Unfortunately, we cannot directly apply Theorem 2.8 to $H_{i}$ and corresponding relative covers, thus we need to consider the groupfication of a neighborhood of the identity in $H_{i}$ .

By Theorem 2.26, there exists $r>0$ such that for all large $i$ , $H_{i}(\tilde{p}_{i},10)$ contains a nilprogression $P_{i}$ which contains $H_{i}(\tilde{p}_{i},4r)$ .

Lemma 6.5.

For any fixed $0<\epsilon<4r$ and sufficiently large $i$ , the groupfication of $H_{i}(\tilde{p}_{i},4r)$ is naturally isomorphic to the groupfication of $H_{i}(\tilde{p}_{i},\epsilon)$ .

Proof.

We have $H_{i}(\tilde{p}_{i},\epsilon)\subset H_{i}(\tilde{p}_{i},4r)\subset P_{i}$ . $H_{i}$ can be generated by $H_{i}(\tilde{p}_{i},\epsilon_{i})$ where $\epsilon_{i}\to 0$ . Using the escape norm, the generators of $P_{i}$ are $u_{j}\in H_{i}(\tilde{p}_{i},\epsilon_{i})$ and the relations in $P_{i}$ are contained in $H_{i}(\tilde{p}_{i},\epsilon)$ . Thus the groupfication of $H_{i}(\tilde{p}_{i},4r)$ or $H_{i}(\tilde{p}_{i},\epsilon)$ is naturally isomorphic to the groupfication of $P_{i}$ which is a lattice in a simply connected nilpotent Lie group. ∎

Let $\hat{H}_{i}$ be the groupfication of $H_{i}(\tilde{p}_{i},4r)$ . For sufficiently large $i$ , we know that $\langle H_{i}(\tilde{p}_{i},r)\rangle=H_{i}$ , thus there is a natural surjective homomorphism $s_{i}:\hat{H}_{i}\to H_{i}$ . We shall use the argument in the proof of Theorem A for $\hat{H}_{i}$ .

We construct a gluing space $\hat{H}_{i}\times_{H_{i}(\tilde{p}_{i},4r)}\bar{B}^{r_{i}}_{2r}(\tilde{p}_{i})$ which is a covering space of $\bar{B}^{r_{i}}_{2r}(p_{i})$ . Let $Y_{i}$ be the pre-image of $\bar{B}^{r_{i}}_{r}(p_{i})$ in $\hat{H}_{i}\times_{H_{i}(\tilde{p}_{i},4r)}\bar{B}^{r_{i}}_{2r}(\tilde{p}_{i})$ . Then $Y_{i}$ is connected since the generators of $\hat{H}_{i}$ is contained in the image of natural injective pseudo-group homomorphism

H_{i}(\tilde{p}_{i},\epsilon_{i})\hookrightarrow\hat{H}_{i}

and $\epsilon_{i}<<r$ when $i$ is large enough.

Thus $H_{i}=\hat{H}_{i}/\mathrm{Ker}(s_{i})$ and define $Z_{i}=Y_{i}/\mathrm{Ker}(s_{i})$ . $Z_{i}$ is the union of all $H_{i}$ -orbits of $\bar{B}^{r_{i}}_{r}(p_{i}^{\prime})$ , and we have

Z_{i}=\pi^{-1}(\bar{B}^{r_{i}}_{r}(p_{i}))\subset r_{i}\widetilde{B}(p_{i},\epsilon_{0},1).

Using 6.3 (and forget $H_{i}^{\prime}$ for now) and the pre-compactness theorem, we obtain the following:

(6.4)

\displaystyle\begin{CD}(Y_{i},\hat{p}_{i},\mathrm{Ker}(s_{i}),\hat{H}_{i})@>{eGH}>{}>(Y,\hat{p},S,\hat{H})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (Z_{i},\tilde{p}_{i},H_{i}=\hat{H}_{i}/\mathrm{Ker}(s_{i}))@>{eGH}>{}>(Y/S=\bar{B}_{r}(0^{k})\times\mathbb{R}^{N-k},0^{N},H=\mathbb{R}^{N-k})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (\bar{B}^{r_{i}}_{r}(p_{i}^{\prime}),p_{i}^{\prime})@>{F_{i}}>{}>(\bar{B}_{r}(0^{k}),0^{k})\end{CD}

where $S$ is the limit group of $\mathrm{Ker}(s_{i})$ , and $H=\hat{H}/S$ .

Lemma 6.6.

$S$ is a trivial group. Thus $Y=\bar{B}^{r_{i}}_{r}(0^{k})\times\mathbb{R}^{N-k}$ and $\hat{H}=\mathbb{R}^{N-k}$ are free translation actions.

Proof.

By [32, 44], for any $g\in\mathrm{Ker}(s_{i})$ , $r_{i}d_{i}(g\hat{p}_{i},\hat{p}_{i})\geq r$ . Thus the limit group $S$ is a discrete group.

We now claim that $S$ admits no isotropy subgroup. Otherwise assume that $\mathrm{id}\neq h\in S$ and $hx=x$ for some $x\in Y$ . We may assume that $x$ is not a boundary point, otherwise we take $\frac{3r}{2}$ -ball instead of $r$ -ball when we define $Y_{i}$ , then $x$ is not on the boundary. Let $\bar{x}$ be the image of $x$ in $Y/S$ . The tangent cone of $\bar{x}$ is $\mathbb{R}^{N}$ , thus tangent cone at $x$ is also $\mathbb{R}^{N}$ . However, $\langle g\rangle$ has non-trivial limit group actions on the tangent cone of $x$ due to volume convergence, thus $\bar{x}$ and $x$ cannot have the same tangent cone, a contradiction.

Thus $S$ are free and discrete isometric group actions. Since $\bar{B}_{r}(0^{k})\times\mathbb{R}^{N-k}$ is simply connected, $S$ is trivial. ∎

Now we have $\hat{H}=H=\mathbb{R}^{N-k}$ and the following diagram:

(6.5)

\displaystyle\begin{CD}(Y_{i},\hat{p}_{i},\mathrm{Ker}(s_{i}),\hat{H}_{i})@>{eGH}>{}>(Y=\bar{B}_{r}(0^{k})\times\mathbb{R}^{N-k},0^{N},\mathrm{id},\hat{H}=\mathbb{R}^{N-k})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (Z_{i},\tilde{p}_{i},H_{i}=\hat{H}_{i}/\mathrm{Ker}(s_{i}))@>{eGH}>{}>(\bar{B}_{r}(0^{k})\times\mathbb{R}^{N-k},0^{N},H=\mathbb{R}^{N-k})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (\bar{B}^{r_{i}}_{r}(p_{i}^{\prime}),p_{i}^{\prime})@>{F_{i}}>{}>(\bar{B}_{r}(0^{k}),0^{k})\end{CD}

Proof of Lemma 6.3.

By Theorem 2.26 and the diagram 6.5, there exists $\epsilon>0$ , for sufficiently large $i$ , we can find a nilporgression $P_{i}^{\prime}\subset\hat{H}_{i}(\hat{p}_{i},1)$ that contains $\hat{H}_{i}(\hat{p}_{i},\epsilon)=\{g\in\hat{H}_{i}|r_{i}d_{i}(g\hat{p},\hat{p})\leq\epsilon\}$ . We may assume $\epsilon<4r$ , then the map

s_{i}:H_{i}(\tilde{p}_{i},\epsilon)\to\hat{H}_{i}(\hat{p}_{i},\epsilon)

is injective [32, 44], thus a pseudo-group isomorphism. Since the groupfication of $H_{i}(\tilde{p}_{i},\epsilon)$ is isomorphic to $\hat{H}_{i}$ by Lemma 6.5, thus $\hat{H}_{i}$ is isomorphic to the groupfication of $\hat{H}_{i}(\hat{p}_{i},\epsilon)$ , which is the groupfication of $P_{i}^{\prime}$ .

We can construct a $(N-k)$ -dim nilpotent group $\mathcal{N}_{i}$ using the nilprogession $P_{i}^{\prime}$ , and endow $N_{i}$ a left-invariant metric as in Lemma 4.5. Then $\hat{H}_{i}$ is isomorphic to a lattice in $\mathcal{N}_{i}$ .

Take $(0^{k},e)\in\bar{B}_{r}(0^{k})\times\mathcal{N}_{i}$ . By Lemma 4.6, we can construct a map

h_{i}^{\prime}:B^{r_{i}}_{r}(\hat{p}_{i})\to B_{r}(0^{k},e)

which is an $\epsilon_{i}$ -GHA and $\epsilon_{i}$ -almost $\hat{H}_{i}(\hat{p}_{i},r)$ -equivariant with some $\epsilon_{i}\to 0$ .

Then by Lemma 4.7, we can construct a global map, possibly dropping points near the boundary if necessary,

h_{i}:(Y_{i},\hat{p}_{i})\to\bar{B}_{r}(0^{k})\times\mathcal{N}_{i}

which is an $\epsilon_{i}$ -GHA on any $r$ -ball and $h$ is $\epsilon_{i}$ -almost $\hat{H}_{i}$ -equivariant.

Now we can find a normal subgroup $\hat{H}_{i}^{\prime}$ in $\hat{H}_{i}$ of finite index, with $\hat{H}_{i}^{\prime}\cap B_{r}(e)=\emptyset$ . Then by Theorem 3.5, after sufficiently blowing up the metric and dropping points near the boundary if necessary, there exists a $\hat{H}_{i}/\hat{H}_{i}^{\prime}$ -equivariant $\epsilon_{i}$ -GHA

f_{\hat{H}_{i}/\hat{H}_{i}^{\prime}}:Y_{i}/\hat{H}_{i}^{\prime}\longrightarrow\bar{B}_{r}(0^{k})\times\mathcal{N}_{i}/\hat{H}_{i}^{\prime},

which is locally almost $N$ -splitting. In particular, $f_{\hat{H}_{i}/\hat{H}_{i}^{\prime}}$ is biHölder. Thus $Y_{i}/H_{i}=\bar{B}^{r_{i}}_{r}(p_{i}^{\prime})$ is biHölder homeomorphic to $\bar{B}_{r}(0^{k})\times\mathcal{N}_{i}/\hat{H}_{i}$ . ∎

6.3. Local relative fundamental groups

Lemma 6.3 is not a proof of Theorem B since $H_{i}$ may not be isomorphic to $\hat{H}_{i}$ ; equivalently, $\mathrm{Ker}(s_{i})$ in 6.5 may not be trivial or the nilpotency rank of $H_{i}$ may be strictly less than $N-k$ . Then the structure of $H_{i}^{\prime}$ is unknown for now. We shall solve this issue by considering another contradiction sequence.

We aim to prove the following in this subsection.

Lemma 6.7.

Take another contradiction sequence if necessary, we may assume that $\hat{H}_{i}=H_{i}$ and $H_{i}$ is a normal subgroup of $H_{i}^{\prime}$ with index $\leq C(N)$ .

Now we have the following from 6.5,

(6.6)

\displaystyle\begin{CD}(Y_{i},\hat{p}_{i},\mathrm{Ker}(s_{i}),\hat{H}_{i})@>{eGH}>{}>(Y=\bar{B}_{r}(0^{k})\times\mathbb{R}^{N-k},0^{N},\mathrm{id},\hat{H}=\mathbb{R}^{N-k})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (Z_{i},\tilde{p}_{i},H_{i}=\hat{H}_{i}/\mathrm{Ker}(s_{i}))@>{GH}>{}>(\bar{B}_{r}(0^{k})\times\mathbb{R}^{N-k},0^{N},H=\mathbb{R}^{N-k})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (\bar{B}^{r_{i}}_{r}(p_{i}^{\prime}),p_{i}^{\prime})@>{eGH}>{}>(\bar{B}_{r}(0^{k}),0^{k}),\end{CD}

and $B^{r_{i}}_{r}(p_{i}^{\prime})$ is biHölder homeomorphic to $B_{r}(0^{k})\times\mathcal{N}_{i}/\hat{H}_{i}$ .

Let $C$ be constant in the generalized Margulis lemma. Consider the quotient space $r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/G_{i}$ which is a finite cover of $r_{i}B_{\epsilon_{0}}(p_{i})$ of index $\leq C$ . For any $g\in G_{i}$ with $g\notin H_{i}$ , $r_{i}d_{i}(g\tilde{p}_{i},\tilde{p}_{i})\geq\frac{1}{\epsilon_{i}}$ . Thus $r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/G_{i}$ is isometric to $r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/H_{i}$ on a large ball. Therefore, for any fixed $s>0$ and $i$ large enough, a connected component of the pre-image of $B_{s}^{r_{i}}(p_{i})$ in $r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/H_{i}$ is a cover of $B_{s}^{r_{i}}(p_{i})$ with index $\leq C$ ; in particular, this cover is contained in a $Cs$ -ball.

Let $\widetilde{H}_{i}^{\prime}$ be the image of natural map $\pi_{1}(B^{r_{i}}_{\frac{r}{20C^{2}}}(p_{i}),p_{i})\to\pi_{1}(B^{r_{i}}_{\frac{r}{10C}}(p_{i}),p_{i})$ . Let $\widetilde{Z}_{i}$ be a connected component of the pre-image of $B^{r_{i}}_{\frac{r}{20C^{2}}}(p_{i})$ in $r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/H_{i}$ . Then $\widetilde{Z}_{i}$ is a finite cover of $B^{r_{i}}_{\frac{r}{20C^{2}}}(p_{i})$ with index at most $C$ . In particular, $\widetilde{Z}_{i}\subset B^{r_{i}}_{\frac{r}{20C}}(p_{i}^{\prime})\subset Z_{i}$ .

Lemma 6.8.

There exists a natural injective homomorphism $i:\hat{H}_{i}\to\widetilde{H}_{i}^{\prime}$ .

Proof.

By Lemma 6.3, $B^{r_{i}}_{r}(p_{i}^{\prime})$ is biHölder homeomorphic to $B^{r_{i}}_{r}(0^{k})\times\mathcal{N}_{i}/\hat{H}_{i}$ and the diameter of $\mathcal{N}_{i}/\hat{H}_{i}<\epsilon_{i}\to 0$ . In particular, the image of

\pi_{1}(B^{r_{i}}_{2\epsilon_{i}}(p_{i}^{\prime}),p_{i}^{\prime})\to\pi_{1}(B^{r_{i}}_{\frac{r}{10}}(p_{i}^{\prime}),p_{i}^{\prime})

is isomorphic to $\hat{H}_{i}$ .

Now we define the homomorphism $i:\hat{H}_{i}\to\widetilde{H}_{i}^{\prime}$ as follows: for any $g\in\hat{H}_{i}$ , we can take a loop $\gamma_{g}\subset B^{r_{i}}_{2\epsilon_{i}}(p_{i}^{\prime})$ at $p_{i}^{\prime}$ that represents $g$ in the image of

\pi_{1}(B^{r_{i}}_{2\epsilon_{i}}(p_{i}^{\prime}),p_{i}^{\prime})\to\pi_{1}(B^{r_{i}}_{\frac{r}{10}}(p_{i}^{\prime}),p_{i}^{\prime}).

Let $\pi:B^{r_{i}}_{2\epsilon_{i}}(p_{i}^{\prime})\to B^{r_{i}}_{2\epsilon_{i}}(p_{i})$ be the projection map and define $i(g)$ to be the element in $\widetilde{H}_{i}^{\prime}$ represented by the loop $\pi(\gamma_{g})$ .

The map $i$ is well-defined because that if two loops $\gamma_{1},\gamma_{2}\subset B^{r_{i}}_{2\epsilon_{i}}(p_{i}^{\prime})$ are homotpic to each other in $B^{r_{i}}_{\frac{r}{10}}(p_{i}^{\prime})$ , then they must be homotopic to each other in $B^{r_{i}}_{\frac{r}{10C}}(p_{i}^{\prime})$ by the local product structure in Lemma 6.3. Thus $\pi(\gamma_{1})$ is homotopic to $\pi(\gamma_{2})$ in $B^{r_{i}}_{\frac{r}{10C}}(p_{i})$ . Then they represent the same element in $\widetilde{H}_{i}$ . Thus $i$ is well-defined. $i$ is a homomorphism by the same argument.

Then we prove that $i$ is injective. Assume that there exists an element $g\in\hat{H}_{i}$ represented by a loop $\gamma_{g}\subset B_{2\epsilon_{i}}(p_{i}^{\prime})$ while $\pi(\gamma_{g})$ is contractible in $B^{r_{i}}_{\frac{r}{10C}}(p_{i})$ . A connected component of the pre-image $B^{r_{i}}_{\frac{r}{10C}}(p_{i})$ is contained in $B^{r_{i}}_{\frac{r}{10}}(p_{i}^{\prime})$ . By the homotopy lifting property, $\gamma_{g}$ is contractible in $B^{r_{i}}_{\frac{r}{10}}(p_{i}^{\prime})$ , thus $g$ is the identity element in the image of $\pi_{1}(B^{r_{i}}_{2\epsilon_{i}}(p_{i}^{\prime}),p_{i}^{\prime})\to\pi_{1}(B^{r_{i}}_{\frac{r}{10}}(p_{i}^{\prime}),p_{i}^{\prime})$ . Then $i$ is injective. ∎

Lemma 6.9.

The index $|\widetilde{H}_{i}^{\prime}:i(\hat{H}_{i})|\leq C$ where $C$ is the constant in the generalized Margulis lemma.

Proof.

Recall that $\widetilde{Z}_{i}$ is a finite cover of $B^{r_{i}}_{\frac{r}{20C^{2}}}(p_{i})$ with index at most $C$ . Assume that $|\widetilde{H}_{i}^{\prime}:i(\hat{H}_{i})|>C$ . Then we can find two loops $\gamma_{1},\gamma_{2}$ in $B^{r_{i}}_{\frac{r}{20C^{2}}}(p_{i})$ at $p_{i}$ , so that $\gamma_{1}\gamma_{2}^{-1}$ does not represent an element in $i(\hat{H}_{i})$ while we lift $\gamma_{1}$ and $\gamma_{2}$ in $\widetilde{Z}_{i}\subset B_{\frac{r}{20C}}(p_{i}^{\prime})$ at $p_{i}^{\prime}$ , saying the $\gamma_{1}^{\prime}$ and $\gamma_{2}^{\prime}$ , then $\gamma_{1}^{\prime}$ and $\gamma_{2}^{\prime}$ have the same endpoint in $\widetilde{Z}_{i}$ .

Since $B^{r_{i}}_{r}(p_{i}^{\prime})$ is biHölder homeomorphic to $B_{r}(0^{k})\times\mathcal{N}_{i}/\hat{H}_{i}$ , $\gamma_{1}^{\prime}(\gamma_{2}^{\prime})^{-1}$ is homotopic to a loop $\gamma_{g}$ corresponding to some $g\in\hat{H}_{i}$ in $B_{r}(0^{k})\times\mathcal{N}_{i}/\hat{H}_{i}$ . Moreover, the homotopy image is contained in $B^{r_{i}}_{\frac{r}{10C}}(p_{i}^{\prime})$ and $\gamma_{g}\subset B_{2\epsilon_{i}}(p_{i}^{\prime})$ as the diameter of $\mathcal{N}_{i}/\hat{H}_{i}$ converges to $0$ . We can project this homotopy map to $B^{r_{i}}_{\frac{r}{10C}}(p_{i})$ , then $\gamma_{1}\gamma_{2}^{-1}$ represents $i(g)\in i(\hat{H}_{i})$ , a contradiction. ∎

Since $\widetilde{H}_{i}^{\prime}$ contains a nilpotent subgroup $i(\hat{H}_{i})$ of index $\leq C$ , then by Lemma 4.8, $\widetilde{H}_{i}^{\prime}$ contains a normal nilpotent subgroup $\widetilde{H}_{i}$ of index $\leq C(N)=C!$ . Since $\hat{H}_{i}$ is torsion free nilpotent group with rank $N-k$ and $\widetilde{H}_{i}$ is a normal subgroup of $i(\hat{H}_{i})$ with finite index, thus rank $(\widetilde{H}_{i})=N-k$ .

Proof of Lemma 6.7.

Now we return to the setup of 6.1 in the beginning of this section. Recall that $\widetilde{H}_{i}^{\prime}$ is the image of $\pi_{1}(B^{r_{i}}_{\frac{r}{20C^{2}}}(p_{i}),p_{i})\to\pi_{1}(B^{r_{i}}_{\frac{r}{10C}}(p_{i}),p_{i})$ . We may take $r_{i}^{\prime}\to\infty$ slowly, then we have another contradiction sequence,

(6.7)

\displaystyle\begin{CD}(r_{i}^{\prime}r_{i}\widetilde{B}(p_{i},\epsilon_{0},1),\tilde{p}_{i},\widetilde{H}_{i},\widetilde{H}_{i}^{\prime})@>{eGH}>{}>(\mathbb{R}^{N},0^{N},G,G^{\prime})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (r_{i}^{\prime}r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/\widetilde{H}_{i},p_{i}^{\prime},\widetilde{H}_{i}^{\prime}/\widetilde{H}_{i})@>{GH}>{}>(Y^{\prime},p^{\prime},\bar{G})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (r_{i}^{\prime}r_{i}B_{\epsilon_{0}}(p_{i}),p_{i})@>{F_{i}}>{}>(\mathbb{R}^{k},0^{k})\end{CD}

The reason that we can replace $G_{i}^{\prime}$ by $\widetilde{H}_{i}^{\prime}$ is that the element not in $\widetilde{H}_{i}^{\prime}$ will disappear in the limit.

Since the index of $\widetilde{H}_{i}$ in $\hat{H}_{i}$ is $\leq C(N)$ , $\widetilde{H}_{i}$ is generated by short elements

\widetilde{H}_{i}(\hat{p}_{i},\epsilon_{i})=\{g\in\widetilde{H}_{i}|r_{i}d_{i}(g\hat{p}_{i},\hat{p}_{i})\leq\epsilon_{i}\}.

Then we can use the same method, for constructing $H_{i},H_{i}^{\prime}$ from the diagram 6.2, to define $\bar{H}_{i}$ and $\bar{H}_{i}^{\prime}$ from the diagram 6.7 using the gap lemma. We may assume $r_{i}^{\prime}\epsilon_{i}\to 0$ , then $\bar{H}_{i}=\widetilde{H}_{i}$ . Since $\bar{H}_{i}^{\prime}$ is a subgroup of $\widetilde{H}_{i}^{\prime}$ , $\bar{H}_{i}$ is a normal subgroup of $\bar{H}_{i}^{\prime}$ of index $\leq C(N)$ .

Then by the same construction of $\hat{H}_{i}$ for $H_{i}$ , we can construct the groupfication $\hat{\bar{H}}_{i}$ using a pseudo-group in $\bar{H}_{i}=\widetilde{H}_{i}$ . There is a natural surjective homomorphism $s_{i}:\hat{\bar{H}}_{i}\to\widetilde{H}_{i}$ . The main improvement is that $\mathrm{Ker}(s_{i})$ must be trivial, since the nilpotency rank of both $\hat{\bar{H}}_{i}$ and $\hat{H}_{i}$ are $N-k$ . Therefore $\widetilde{H}_{i}$ is isomorphic to the groupfication $\hat{\bar{H}}_{i}$ .

Working on the diagram 6.7 with $\widetilde{H}_{i}=\bar{H}_{i}$ and $\bar{H}_{i}^{\prime}$ if necessary, we may assume that $H_{i}$ is normal subgroup of $H_{i}^{\prime}$ of index $\leq C(N)$ and $\hat{H}_{i}=H_{i}$ . ∎

6.4. Proof of Theorem B

Now we can prove that $f_{i}:X_{i}\to K$ is a fibration map. We summarize the differences in this subsection compared with the proof of Lemma 6.3. The first difference is that, since $H_{i}$ is isomorphic to the lattice by Lemma 6.7, $H_{i}^{\prime}$ can be identified as a discrete subset of $\mathcal{N}_{i}\rtimes\text{Aut}(\mathcal{N}_{i})$ . Then we can construct an infranil-manifold fiber on $X_{i}$ . The second difference is that we apply the gluing argument from Theorem 3.5 carefully so that $F_{i}=f_{i}\circ\mathrm{exp}_{p}^{-1}$ is the a fibration map near $p$ , thus $f_{i}$ is a fibration map.

Proof of Theorem B, the existence of the fibration with an infranil-manifold fiber.

Consider the diagram 6.3

$\begin{CD}(r_{i}\widetilde{B}(p_{i},\epsilon_{0},1),\tilde{p}_{i},H_{i},H_{i}^{\prime})@>{eGH}>{}>(\mathbb{R}^{N},0^{N},H=\mathbb{R}^{N-k},H^{\prime})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)/H_{i}^{\prime},p_{i}^{\prime},H_{i}/H_{i}^{\prime})@>{GH}>{}>(\mathbb{R}^{k},0^{k},\mathrm{id})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (r_{i}B_{\epsilon_{0}}(p_{i}),p_{i})@>{F_{i}}>{}>(\mathbb{R}^{k},0^{k}).\end{CD}$

We assumed that $F_{i}=\mathrm{exp}_{p}^{-1}\circ f_{i}$ is not a fibration around $p$ . The goal is to show that $F_{i}$ is a fibration when $i$ is large enough, thus a contradiction.

Now we apply Lemma 6.7 to the diagram 6.6,

(6.8)

\displaystyle\begin{CD}(Z_{i},\tilde{p}_{i},H_{i}=\hat{H}_{i},H_{i}^{\prime})@>{eGH}>{}>(\bar{B}_{r}(0^{k})\times\mathbb{R}^{N-k},0^{N},H=\mathbb{R}^{N-k},H^{\prime})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (\bar{B}^{r_{i}}_{r}(p_{i}^{\prime}),p_{i}^{\prime},H_{i}/H_{i}^{\prime})@>{eGH}>{}>(\bar{B}_{r}(0^{k}),0^{k},\mathrm{id})\\ @V{}V{\pi}V@V{}V{\pi}V\\ (\bar{B}^{r_{i}}_{r}(p_{i}),p_{i})@>{F_{i}}>{}>(\bar{B}_{r}(0^{k}),0^{k})\end{CD}

By Lemma 6.7, $H_{i}=\hat{H}_{i}$ is a lattice in $\mathcal{N}_{i}$ and $H_{i}^{\prime}$ can be identified as a discrete subset of $\mathcal{N}_{i}\rtimes\text{Aut}(\mathcal{N}_{i})$ . Use the argument in Lemma 4.6 and 4.7, we can construct a global map

\tilde{h}_{i}:Z_{i}\to\bar{B}_{r}(0^{k})\times\mathcal{N}_{i}

is an $\epsilon_{i}$ -GHA on any $r$ -ball and $\epsilon_{i}$ -almost $H_{i}^{\prime}$ -equivariant.

Now we can find a normal subgroup $H_{i}^{\prime\prime}$ in $H_{i}^{\prime}$ with finite index and $B_{r}(e)\cap H_{i}^{\prime\prime}=\emptyset$ where $e\in\mathcal{N}_{i}$ . Then $(Z_{i}/H_{i}^{\prime\prime},H_{i}^{\prime}/H_{i}^{\prime\prime})$ is eGH close to $(\bar{B}_{r}(0^{k})\times\mathcal{N}_{i}/H_{i}^{\prime\prime},H_{i}^{\prime}/H_{i}^{\prime\prime})$ on any $r$ -ball. We use Theorem 3.5 to construct a $H_{i}^{\prime}/H_{i}^{\prime\prime}$ -equivariant homeomorphism from $Z_{i}/H_{i}^{\prime\prime}$ to $\bar{B}_{r}(0^{k})\times\mathcal{N}_{i}/H_{i}^{\prime\prime}$ , dropping some points near the boundary if necessary. Thus $Z_{i}/H_{i}^{\prime}=\bar{B}^{r_{i}}_{r}(p_{i})$ , dropping some points near the boundary if necessary, is biHölder homeomorphic to $\bar{B}_{r}(0^{k})\times N_{i}/H_{i}^{\prime}$ .

Recall that

\tilde{F}_{i}:r_{i}\widetilde{B}(p_{i},\epsilon_{0},1)\overset{\pi}{\longrightarrow}r_{i}B_{\epsilon_{i}}(p_{i})\overset{F_{i}}{\longrightarrow}(\mathbb{R}^{k},0^{k}),

is almost $k$ -splitting. We need to use Theorem 3.5 more carefully to prove that $F_{i}$ is exactly the fibration map. The construction in Theorem 3.5 is to glue local almost $N$ -splitting maps and the group action orbits. Every time we choose local almost $N$ -splitting map on $Z_{i}/H_{i}^{\prime\prime}$ , we take first $\mathbb{R}^{k}$ -component exactly to be $\tilde{F}_{i}$ . Then the gluing by the center of mass keeps the value on the first $\mathbb{R}^{k}$ component, since different local almost $N$ -splitting or $H_{i}^{\prime}$ orbits have the same $\tilde{F}_{i}$ value at a point. Therefore the gluing only changes the value on the $\mathcal{N}_{i}$ component.

In particular, $F_{i}:B_{r}(p_{i})\to\mathbb{R}^{k}$ is the exactly $\mathbb{R}^{k}$ component of the biHölder homeomorphic from $B_{r}(p_{i})$ to $B_{r}(0^{k})\times\mathcal{N}_{i}/H_{i}^{\prime}$ constructed above,

F_{i}:B_{r}(p_{i})\overset{\mathrm{homeo}}{\longrightarrow}B_{r}(0^{k})\times\mathcal{N}_{i}/H_{i}^{\prime}\overset{\pi}{\longrightarrow}B_{r}(0^{k}).

Therefore $F_{i}$ is a fibration map with the infranil-manifold fiber. ∎

We next show that nilpotent structure of $\mathcal{N}_{i}$ , as described in Theorem B, does not depend on the choice of base point on $X_{i}$ , which implies that the structure group in Theorem B can be affine; we refer to the gluing arguments in [9, 35] for further details.

Recall that nilpotent structure of $\mathcal{N}_{i}$ in Theorem B is determined by the escape norm on $H_{i}(\tilde{p}_{i},r)$ . Let $A=H_{i}(\tilde{p},r)$ , then $\mathcal{N}_{i}$ is constructed from a nilprogression $P(u_{1},...,u_{N-k};C_{1},...,C_{N-k})$ with the escape norm $||u_{1}||_{A}\leq||u_{2}||_{A}\leq...\leq||u_{N-k}||_{A}$ . And $H_{i}=\hat{H}_{i}$ is a lattice of $\mathcal{N}_{i}$ .

Passing to a subsequence if necessary, we may assume that $\frac{||u_{j+1}||_{A}}{||u_{j}||_{A}}$ converges to a real number or $\infty$ as $i\to\infty$ . Thus we can find $K$ and $1=l_{1}<l_{2}<...<l_{K}\leq N-k$ such that the following two conditions holds for sufficiently large $i$ :

\frac{||u_{j}||_{A}}{||u_{l}||_{A}}\leq C,\ \mathrm{if\ for\ some\ }s,l_{s}\leq l<j<l_{s+1},

and

\frac{||u_{j}||_{A}}{||u_{l}||_{A}}\to\infty,\ \mathrm{if\ for\ some\ s,}\ l<l_{s}\leq j,

as $i\to\infty$ .

Fix a large $i$ , define $G_{s}=\langle u_{1},...,u_{l_{s}-1}\rangle\subset H_{i}$ , $1\leq s\leq K$ , then $G_{s}/G_{s-1}$ is in the center of $H_{i}/G_{s-1}$ due to Theorem 2.19 and the construction of the nilprogression. Let $\mathcal{N}_{i,s}$ be the simply connected subgroup of $\mathcal{N}_{i}$ with lattice $G_{s}$ . Thus $(\mathcal{N}_{i}/\mathcal{N}_{i,s-1})/(G_{s}/G_{s-1})$ is a torus bundle over $\mathcal{N}_{i}/\mathcal{N}_{i,s}$ .

For sufficiently large $i$ , we shall show that the constructions of $\mathcal{N}_{i}$ and $G_{s}$ above are independent of the choice of base point $p_{i}\in X_{i}$ . This will allow us to use a gluing argument in [9, 35] to modify the fibration map and reduce the structure group in Theorem B to be affine.

Consider 6.3, assume $\tilde{x}_{i},\tilde{y}_{i}\in B^{r_{i}}_{1}(\tilde{p}_{i})$ , for any $0<r<1/100$ , define

A_{1}=\{g\in H_{i}|r_{i}d_{i}(g,g\tilde{x}_{i})\leq r\},

A_{2}=\{g\in H_{i}|r_{i}d_{i}(g,g\tilde{y}_{i})\leq r\}.

We need to show that for any $g\in A_{1}\cap A_{2}$ and sufficiently large $i$ , $\frac{1}{2}\leq\frac{||g||_{A_{1}}}{||g||_{A_{2}}}\leq 2$ , which implies that the escape norms by $A_{1}$ and $A_{2}$ give the same nilpotent structure of $H_{i}$ .

By the definition of escape norm, $g^{1/||g||_{A_{1}}}\notin A_{1}$ . Then $d(g^{1/||g||_{A_{1}}}\tilde{x}_{i},\tilde{x}_{i})\geq r$ . Since the limit of $H_{i}$ is a free translation group $\mathbb{R}^{N-k}$ by Lemma 6.1, $d(g^{1/||g||_{A_{1}}}\tilde{y}_{i},\tilde{y})\geq\frac{2r}{3}$ . Then $d(g^{2/||g||_{A_{1}}}\tilde{y}_{i},)\geq r$ . In particular, $||g||_{A_{2}}\geq\frac{||g||_{A_{1}}}{2}$ . Similarly $||g||_{A_{1}}\geq\frac{||g||_{A_{2}}}{2}$ . Thus we have proved that the nilpotent structure of the fiber does not depend on the choice of base point in a small neighborhood of $p_{i}$ . Then by a connectedness argument, we conclude that the nilpotent structure of the fiber is independent of the choice of base point on $X_{i}$ for sufficiently large $i$ , thus the structure group in Theorem B is affine.

7. Limit of RCD $(N-1,N)$ spaces with bounded covering geometry

In this section we consider a sequence of pointed RCD $(-(N-1),N)$ spaces $(X_{i},d_{i},\mathcal{H}^{N},p_{i})$ with $(1,v)$ -bound covering geometry and assume the following convergence:

(7.1)

(X_{i},d_{i},\mathcal{H}^{N},p_{i})\overset{\mathrm{pmGH}}{\longrightarrow}(X,d,\mathfrak{m},p).

Assume that the rectifiable dimension of $X$ is $k$ . We want to show that any $k$ -regular point in $X$ is a manifold point.

Theorem 7.1.

In the context of 7.1, assume that $x\in X$ is a $k$ -regular point. Then there exists a neighborhood of $x$ which is biHölder homeomorphic to an open ball in $\mathbb{R}^{k}$ .

Let $x_{i}\in X_{i}$ converge to $x\in X$ . Let $G_{i}$ be the image of natural homomoephism

\pi_{1}(B_{\frac{1}{2}}(x_{i}),x_{i})\to\pi_{1}(B_{1}(x_{i}),x_{i}).

Then let $\widetilde{B}(x_{i},\frac{1}{2},1)$ be a connected component of the pre-image of $B_{\frac{1}{2}}(x_{i})$ in the universal cover of $B_{1}(x_{i})$ . Take $\tilde{x}_{i}\in\widetilde{B}(x_{i},\frac{1}{2},1)$ as a lift of $x_{i}$ . By the pre-compactness theorem 2.9, passing to a subsequence if necessary,

(7.2)

\displaystyle\begin{CD}(\widetilde{B}(x_{i},\frac{1}{2},1),\tilde{p}_{i},G_{i})@>{eGH}>{}>(Y,\tilde{x},G)\\ @V{}V{\pi}V@V{}V{\pi}V\\ (B_{\frac{1}{2}}(x_{i}),x_{i})@>{GH}>{}>(\bar{B}_{\frac{1}{2}}(x),x)\end{CD}

Since $x$ is a $k$ -regular point, $\tilde{x}$ must be a $N$ -regular point by [23].

We next show that an almost $k$ -splitting map near $x$ can be lifted to an almost $k$ -splitting map near $\tilde{x}$ .

Lemma 7.2.

In the context of 7.1, there exists $C(N)>0$ so that the following holds. Assume that there exist small $\epsilon,r>0$ , and

f:B_{r}(x)\to\mathbb{R}^{k}

is harmonic and $(k,\epsilon)$ -splitting. Define the map

\tilde{f}:B_{\frac{r}{2}}(\tilde{x})\to\mathbb{R}^{k},\ \tilde{f}=f\circ\pi.

Then $\tilde{f}$ is harmonic and $(k,C\epsilon)$ -splitting.

Proof.

By Corollary 4.12 in [1], we can find harmonic maps

f_{i}:B_{\frac{2r}{3}}(x_{i})\to\mathbb{R}^{k}

converging to $f_{|B_{\frac{2r}{3}}(x_{i})}$ in the $H^{1,2}$ sense. In particular, $f_{i}$ is $(k,2\epsilon)$ -splitting for large $i$ .

Define

\tilde{f}_{i}:B_{\frac{2r}{3}}(\tilde{x}_{i})\to\mathbb{R}^{k},\ \tilde{f}_{i}=f_{i}\circ\pi.

Then $\tilde{f}_{i}$ is a harmonic and $(k,C\epsilon)$ -splitting by the covering lemma 2.12. Passing to a subsequence if necessary, by Theorem 4.4 in [1], $\tilde{f}_{i}$ has a $H^{1,2}$ limit $\tilde{g}$ which is harmonic on $B_{\frac{r}{2}}(\tilde{x})$ . Then $\tilde{g}$ is also $(k,C\epsilon)$ -splitting. By the equivariant convergence, $f=\tilde{g}\circ\pi$ on $B_{\frac{r}{2}}(x)$ , thus $\tilde{f}=\tilde{g}$ is harmonic and $(k,C\epsilon)$ -splitting. ∎

We prove that the geometric transformation theorem holds at regular points in $X$ .

Lemma 7.3.

In the context of 7.1, for any $\delta>0$ , there exists $\epsilon>0$ so that the following holds. Assume that there exists $r_{0}\leq 1$ and a $(k,\epsilon)$ -splitting map

f:B_{r_{0}}(x)\to\mathbb{R}^{k}.

Then for any $s\leq\frac{r_{0}}{10}$ , there exists an $k\times k$ lower triangular matrix $T_{s}$ such that

T_{s}(f):B_{s}(x)\to\mathbb{R}^{k}

is a $(k,\delta)$ -splitting and $|T_{s}|\leq s^{-\delta}$ .

Proof.

Take a small $\epsilon>0$ to be decided later. Since $x$ is regular, there exists $r_{0}>0$ and a $(k,\epsilon)$ -splitting map

f:B_{r_{0}}(x)\to\mathbb{R}^{k}.

Then $\tilde{f}=f\circ\pi$ is $(k,C\epsilon)$ -splitting by Lemma 7.2.

The rectifiable dimension of $X$ is $k$ . By [29], any tangent cone of $X$ can not split an $\mathbb{R}^{k+1}$ factor. In particular, $B_{r_{0}}(x)$ is $r_{0}\Phi(\epsilon|k,N)$ -close to a $r_{0}$ -ball in $\mathbb{R}^{k}$ . By [23], $B_{r_{0}}(\tilde{x})$ is $r_{0}\Phi(\epsilon|k,N,v)$ -close to $r_{0}$ -ball in $\mathbb{R}^{N}$ .

Now we can find

\tilde{h}:B_{\frac{r_{0}}{2}}(\tilde{x})\to\mathbb{R}^{N-k}

such that the pair $(\tilde{f},\tilde{h}):B_{\frac{r_{0}}{2}}(\tilde{x})\to\mathbb{R}^{N}$ forms a $(N,\Phi(\epsilon|k,N,v))$ -splitting map. Apply the transformation theorem 2.6 to $(\tilde{f},\tilde{h})$ . When $\epsilon$ is small enough, for any $s\leq\frac{r_{0}}{10}$ , we can find a $N\times N$ lower triangular matrix $\tilde{T}_{s}$ such that

\tilde{T}_{s}(\tilde{f},\tilde{h}):B_{3s}(\tilde{x})\to\mathbb{R}^{N}

a $(N,\delta)$ -splitting and $|\tilde{T}_{s}|\leq s^{-\delta}$ .

Take $T_{s}$ to be the upper left $k\times k$ submatrix of $\tilde{T}_{s}$ . Then $T_{s}$ is a $k\times k$ lower triangular matrix and $T_{s}\tilde{f}$ is a $(k,\delta)$ -splitting on $B_{3s}(\tilde{x})$ . By a similar argument in Lemma 7.2, we conclude that $T_{s}f$ is $(N,C\delta)$ -splitting on $B_{s}(x)$ . Moreover, $|T_{s}|\leq|\tilde{T}_{s}|\leq s^{-\delta}$ . ∎

Proof of Theorem 7.1.

The proof of Theorem 7.1 follows from the proof of canonical Reifenberg theorem in [12] and [22]. Take any small $\delta>0$ , by Lemma 7.3, there exists $r_{0}>0$ and a $(k,\epsilon)$ -splitting map

f:B_{4r_{0}}(x)\to\mathbb{R}^{k}

so that for each $y\in B_{2r_{0}}(x)$ and $s\leq r_{0}$ , there exists a $k\times k$ lower triangular matrix $T_{s,y}$ so that

T_{s,y}f:B_{s}(y)\to\mathbb{R}^{k}

is a $(k,\delta)$ -splitting map with $|T_{s,y}|\leq s^{-\delta}$ .

We shall show that $f$ is biHölder from $B_{r_{0}}(x)$ to its image. For any $y_{1},y_{2}\in B_{r_{0}}(x)$ , take $s=d(y_{1},y_{2})$ . Since $T_{s,y_{1}}f:B_{s}(y_{1})\to\mathbb{R}^{k}$ is $(k,\delta)$ -splitting and the dimension of $X$ is $k$ , it must be a $\Phi(\delta|k)s$ -GHA. Thus

d(T_{s,y_{1}}f(y_{1}),T_{s,y_{1}}f(y_{2}))\geq(1-\Phi(\delta|k))d(y_{1},y_{2}).

Since $|T_{s,y_{1}}|\leq s^{-\delta}$ and $s=d(y_{1},y_{2})$ ,

d(f(y_{1}),f(y_{2}))\geq s^{\delta}d(T_{s,y_{1}}f(y_{1}),T_{s,y_{1}}f(y_{2}))\geq(1-\Phi(\delta|k,N))d(y_{1},y_{2})^{1+\delta}.

On the other hand, since $f$ is harmonic, $|\nabla f^{a}|\leq 1+\Phi(\delta|k)$ , for any $a=1,2,...k$ , thus

d(f(y_{1}),f(y_{2}))\leq(1+\Phi(\delta|k))d(y_{1},y_{2}).

The biHölder estimate holds, completing the proof. ∎

Proof of Theorem 1.9.

The part (a) is exactly Theorem 7.1 and the part (b) follows from the construction in Theorem B. ∎

References

[1] L. Ambrosio and S. Honda. Local spectral convergence in RCD $*$ (K,N) spaces. Nonlinear Anal. 177 (2018), 1–23., 2018.
[2] Michael T. Anderson. Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem. Duke Mathematical Journal, 68(1):67 – 82, 1992.
[3] E Breuillard, B Green, and T Tao. The structure of approximate groups. Publ. Math. Inst. Hautes Etudes Sci., 116:115–221, 2012.
[4] E. Brue, A. Naber, and D. Semola. Boundary regularity and stability for spaces with Ricci bounded below. Invent. Math. 228 (2022) no. 2, 777–891, 2022.
[5] E. Brue and D. Semola. Constancy of the dimension in codimension one and locality of the unitnormal on RCD(K,N) spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 24 (2023), no.3, 1765-1816., 2023.
[6] Peter Buser and Hermann Karcher. Gromovs almost flat manifolds. Société mathématique de France, 1981.
[7] Jeff Cheeger and Tobias H. Colding. On the structure of spaces with Ricci curvature bounded below. i. J. Differential Geom., 46(3):406–480, 1997.
[8] Jeff Cheeger and Tobias H. Colding. On the structure of spaces with Ricci curvature bounded below. ii. J. Differential Geom., 54(1):13–35, 2000.
[9] Jeff Cheeger, Kenji Fukaya, and Mikhael Gromov. Nilpotent structures and invariant metrics on collapsed manifolds. Journal of the American Mathematical Society, 5(2):327–372, 1992.
[10] Jeff Cheeger and Mikhael Gromov. Collapsing Riemannian manifolds while keeping their curvature bounded. I. Journal of Differential Geometry, 23(3):309–346, 1986.
[11] Jeff Cheeger and Mikhael Gromov. Collapsing Riemannian manifolds while keeping their curvature bounded. II. Journal of Differential Geometry, 32(1):269– 298, 1990.
[12] Jeff Cheeger, Wenshuai Jiang, and Aaron Naber. Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below. Annals of Mathematics, 193(2):407–538, 2021.
[13] Jeff Cheeger and Aaron Naber. Regularity of Einstein manifolds and the codimension 4 conjecture. Ann. of Math., 182(3):1093–1165, 2015.
[14] Q. Deng, J. Santos-Rodríguez, S. Zamora, and X. Zhao. Margulis Lemma on RCD $(K,N)$ spaces. arXiv preprint, 2023.
[15] Kenji Fukaya. Theory of convergence for riemannian orbifolds. Japan. J. Math. (N.S.), 12(1):121–160, 1986.
[16] Kenji Fukaya. Collapsing Riemannian manifolds to ones of lower dimensions. Journal of Differential Geometry, 25(1):139–156, 1987.
[17] Kenji Fukaya. A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters. Journal of Differential Geometry, 28(1):1–21, 1988.
[18] Kenji Fukaya. Collapsing Riemannian manifolds to ones with lower dimension II. Journal of the Mathematical Society of Japan, 41(2):333–356, 1989.
[19] Kenji Fukaya and Takao Yamaguchi. The fundamental groups of almost nonnegatively curved manifolds. Annals of Mathematics, 136(2):253–333, 1992.
[20] Mikhael Gromov. Almost flat manifolds. J. Diff. Geom., 13(2):231–241, 1978.
[21] Luis Guijarro and Jaime Santos-Rodríguez. On the isometry group of RCD $*$ (K,N)-spaces. manuscripta mathematica, 158:441–461, 2019.
[22] S. Honda and Y. Peng. A note on the topological stability theorem from spaces to Riemannian manifolds. Manuscripta Math. 172 (2023), no.3-4, 971-1007., 2023.
[23] Hongzhi Huang. Fibrations, and stability for compact group actions on manifolds with local bounded Ricci covering geometry. Front. Math. China, 15(1):69–89, 2020.
[24] Hongzhi Huang, Lingling Kong, Xiaochun Rong, and Shicheng Xu. Collapsed manifolds with Ricci bounded covering geometry. Transactions of the American Mathematical Society, 373(11):8039–8057, 2022.
[25] Erik Hupp, Aaron Naber, and Kai-Hsiang Wang. Lower Ricci curvature and nonexistence of manifold structure. arXiv:2308.03909, 2023.
[26] Vitali Kapovitch. Mixed curvature almost flat manifolds. Geom. Topol. 25 (2021) 2017-2059, 2021.
[27] Vitali Kapovitch, Martin Kell, and Christian Ketterer. On the structure of RCD spaces with upper curvature bounds. Mathematische Zeitschrift, 2022, No 4, p. 3469-3502, 2022.
[28] Vitali Kapovitch and Burkhard Wilking. Structure of fundamental groups of manifolds with Ricci curvature bounded below. arXiv:1105.5955, 2011.
[29] Yu Kitabeppu. A sufficient condition to a regular set of positive measure on RCD spaces. Potential Anal. 51 (2019), no. 2, 179–196., 2019.
[30] K.B. Lee and F. Raymond. Rigidity of almost crystallographic groups. Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982), Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 73-78. ISBN:0-8218-5039-3., 1985.
[31] Andrea Mondino and Guofang Wei. On the universal cover and the fundamental group of an RCD $*(K,N)$ -space. J. Reine Angew. Math., 2019(753):211–237, 2019.
[32] Jiayin Pan and Jikang Wang. Some topological results of Ricci limit spaces. Transactions of the American Mathematical Society, 375(12):8445–8464, 2022.
[33] G. De Philippis and N. Gigli. Non-collapsed spaces with Ricci curvature bounded from below. J.Ec. polytech. Math. 5 (2018), 613-650., 2018.
[34] Xiaochun Rong. A new proof of the Gromov’s theorem on almost flat manifolds. arXiv:1906.03377v2, 2019.
[35] Xiaochun Rong. Collapsed manifolds with local ricci bounded covering geometry. arXiv:2211.09998, 2022.
[36] Ernst A. Ruh. Almost flat manifolds. J. Differential Geom., 17(1):1–14, 1982.
[37] Jaime Santos-Rodriguez and Sergio Zamora. On fundamental groups of RCD spaces. Journal für die reine und angewandte Mathematik (Crelles Journal), vol. 2023, no. 799, 2023, pp. 249-286., 2023.
[38] Christina Sormani and Guofang Wei. Universal covers for Hausdorff limits of noncompact spaces. Transactions of the American Mathematical Society, 356(3):1233–1270, 2004.
[39] Gerardo Sosa. The isometry group of an RCD $*$ space is Lie. Potential Analysis, 49:267–286, 2018.
[40] Jikang Wang. On the limit of simply connected manifolds with discrete isometric cocompact group actions. arXiv:2307.07658, 2023.
[41] Jikang Wang. RCD $*$ (K,N) spaces are semi-locally simply connected. Journal für die reine und angewandte Mathematik (Crelles Journal), vol. 2024, no. 806, 2024, pp. 1-7., 2024.
[42] Jikang Wang. Ricci limit spaces are semi-locally simply connected. Journal of Differential Geometry, J. Differential Geom. 128(3), 1301-1314, (November 2024), 2024.
[43] Shicheng Xu. Precompactness of domains with lower Ricci curvature bound under Gromov-Hausdorff topology. arXiv:2311.05140, 2023.
[44] Sergio Zamora. Limits of almost homogeneous spaces and their fundamental groups. Groups Geom. Dyn. 18 (2024), no. 3, pp. 761–798, 2024.
[45] Sergio Zamora and Xingyu Zhu. Topological rigidity of small RCD(K,N) spaces with maximal rank. arXiv:2406.10189, 2024.
[46] Shengxuan Zhou. Examples of Ricci limit spaces with infinite holes. arXiv:2404.00619, 2024.

On the non-collapsed RCD spaces with local bounded covering geometry

Abstract.

1. Introduction

Theorem 1.1 (Gromov’s almost flat manifold theorem, [20, 34, 36]).

Theorem 1.2 (smooth fibration, [16, 9]).

Theorem 1.3 ([24, 35]).

Theorem 1.4.

Remark 1.5.

Theorem A.

Theorem 1.6 ([26]).

Theorem 1.7.

Theorem 1.8 ([23, 35]).

Theorem B.

Theorem 1.9.

Remark 1.10.

Remark 1.11.

2. Preliminaries

2.1. Equivariant Gromov-Hausdorff convergence and isometry group on a Ricci limit space

Theorem 2.1 ([15, 19]).

2.2. Geometric Theory of RCD(K,N)(K,N) spaces

Definition 2.2.

Definition 2.3.

Theorem 2.4 (Generalized Margulis lemma, [14]).

Theorem 2.5 (Volume convergence, [33]).

Theorem 2.6 (Transformation, [4]).

Theorem 2.7 (Canonical Reifenberg method [12, 22]).

2.3. The universal covers and local relatives covers of RCD(K,N)(K,N) spaces

Theorem 2.8.

Theorem 2.9 (Precompactness of relative covers, [43]).

Remark 2.10.

Lemma 2.11.

Lemma 2.12 (Covering lemma, [28]).

Lemma 2.13 (Gap lemma, [28]).

2.4. Approximate groups and almost homogeneous spaces

Definition 2.14.

Definition 2.15.

Definition 2.16.

Definition 2.17.

Definition 2.18.

Theorem 2.19 (escape norm estimate, [3]).

Remark 2.20.

Definition 2.21.

Definition 2.22.

Theorem 2.23 ([3, 44]).

Remark 2.24.

Remark 2.25.

Theorem 2.26 ([3, 44]).

Remark 2.27.

Definition 2.28.

Theorem 2.29.

Theorem 2.30 (Lie algebra structure convergence, [44]).

3. Constructing a GHA map which is locally almost splitting

Theorem 3.1 (smooth fibration, [23]).

Theorem 3.2 (stability for compact group actions, [23]).

Theorem 3.3 (Almost kk-splitting).

Proof.

Remark 3.4.

Theorem 3.5 (GG-equivariant).

Remark 3.6.

Proof.

4. Proof of Theorem A: construct an infranil-manifold

Theorem A.

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Corollary 4.3.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

Lemma 4.6 (Local eGH closeness).

Proof.

Lemma 4.7 (Extend a local approximation to the global map, [40]).

Lemma 4.8.

Proof.

Proof of Theorem A.

Remark 4.9.

5. Mixed curvature and almost flat manifolds theorem

Theorem 5.1.

2.2. Geometric Theory of RCD $(K,N)$ spaces

2.3. The universal covers and local relatives covers of RCD $(K,N)$ spaces

Theorem 3.3 (Almost $k$ -splitting).

Theorem 3.5 ( $G$ -equivariant).

7. Limit of RCD $(N-1,N)$ spaces with bounded covering geometry