Croke-Kleiner admissible groups: Property (QT) and quasiconvexity

A quasi-tree is a geodesic metric space quasi-isometric to a tree. Recently, Bestvina, Bromberg and Fujiwara [BBF19] introduced a (QT) property for a finitely generated group: $G$ acts properly on a finite product of quasi-trees so that the orbital map from $G$ with word metrics is a quasi-isometric embedding. This is a stronger property of the finite asymptotic dimension by recalling that a quasi-isometric embedding implies finite asdim of $G$. It is known that Coxeter groups have property (QT) (see [DJ99]), and thus every right-angled Artin group has property (QT) (see Induction 2.2 in [BBF19]). Furthermore, the fundamental group of a compact special cube complex is undistorted in RAAGS (see [HW08]) and then has property (QT). As a consequence, many 3-manifold groups have property (QT), among which we wish to mention chargeless (including flip) graph manifolds [HP15] and finite volume hyperbolic 3-manifolds [Wis20]. In [BBF19], residually finite hyperbolic groups and mapping class groups are proven to have property (QT). It is natural to ask which other groups have property (QT) rather than these groups above.

The main result of this paper adds flip CKA actions into the list of groups which have property (QT). The notion of an omnipotent group is introduced by Wise in [Wis00] and has found many applications in subgroup separability. We refer the reader to Definition 5.10 for its definition and note here that free groups [Wis00], surfaces groups [Baj07], and the more general class of virtually special hyperbolic groups [Wis20] are omnipotent.

We now give an outline of the proof of Theorem 1.1 and explain some intermediate results, which we believe are of independent interest.

Let us describe briefly the construction of $\mathcal{X}\in\{\mathcal{X}_{1},\mathcal{X}_{2}\}$. By Bass-Serre theory, $G$ acts on the Bass-Serre tree $T$ with vertex groups $G_{v}$ and edge groups $G_{e}$. Let $\mathcal{V}$ be one of the two sets of vertices in $T$ with pairwise even distance. Note that $G_{v}$ is the central extension of a hyperbolic group $H_{v}$ by $\mathbb{Z}$, so acts geometrically on a metric product $Y_{v}=\overline{Y}_{v}\times\mathbb{R}$ where $H_{v}$ acts geometrically on $\overline{Y}_{v}$ and $Z(G_{v})$ acts by translation on $\mathbb{R}$-lines. Roughly, the space $\mathcal{X}$ is obtained by isometric gluing of the boundary lines of $\overline{Y}_{v}$’s over vertices $v$ in the link of every $w\in T^{0}-\mathcal{V}$. In proving Proposition 1.4, the main tool is the construction of a class of quasi-geodesic paths called special paths between any two points in $X$. See Section 3 for the details and related discussion after Theorem 1.6 below.

To endow an action on $\mathcal{X}$, we pass to an index at most 2 subgroup $\dot{G}$ preserving $\mathcal{V}$ and the stabilizer in $\dot{G}$ of $v\in\mathcal{V}$ is $G_{v}$ by Lemma 4.6. Under the assumptions on $G_{v}$ and $G_{e}$’s, $\dot{G}$ acts by isometry on $\mathcal{X}$ and the Q.I. embedding is $\dot{G}$-equivariant.

To prove Theorem 1.1, we exploit the strategy as [BBF19] to produce a proper action on products of quasi-trees. By Lemma 4.17, we first produce enough quasi-lines

for the hyperbolic space $\mathcal{X}$ where $\mathbb{A}_{v}$ is a $H_{v}$-finite set of quasi-lines in $\overline{Y}_{v}$ so that the so-called distance formula follows in Proposition 4.20. On the other hand, the “crowd” quasi-lines in $\mathbb{A}$ may fail to satisfy the projection axioms in [BBF15] with projection constant required from the distance formula. Thus, we have to partition $\mathbb{A}$ into finite sub-collections of sparse quasi-lines: $\forall\gamma\neq\gamma^{\prime}$, $d(\gamma,\gamma^{\prime})>\theta$ for a uniform constant $\theta$.

Using local finiteness, we can partition quasi-lines without respecting group action and prove the following general result. This generalize the results of Hume-Sisto in [HS13] to flip CKA actions.

However, the difficulty in establishing property (QT) is to partition all quasi-lines $\mathbb{A}=\cup_{i=1}^{n}\mathbb{A}_{i}$ so that each $\mathbb{A}_{i}$ is $\dot{G}$-invariant and sparse. In §5.1, we cone off the boundary lines of $Y_{v}$’s so that $\mathbb{A}_{v}$’s from different pieces $Y_{v}$ are “isolated”. The gives the coned-off space $\dot{\mathcal{X}}$ with new distance formula in Proposition 5.9 so that $\mathcal{X}$ is quasi-isometric embedded into the product of $\dot{\mathcal{X}}$ with a quasi-tree from the boundary lines. See Proposition 5.7.

The goal is then to find a finite index subgroup $\ddot{G}<\dot{G}<G$ so that each orbit in $\mathbb{A}$ is sparse. This is done in the following two steps:

By residual finiteness of $H_{v}$, we first find a finite index subgroup $K_{v}<H_{v}$ whose orbit in $\mathbb{A}_{v}$ is sparse. This follows the same argument in [BBF19]. Secondly, we need to reassemble those finite index subgroups $K_{v}$ as a finite index group $\ddot{G}$ so that the orbit in $\mathbb{A}$ is sparse. This step uses crucially the omnipotence, with details given in §5.4. The projection axioms thus fulfilled for each $\ddot{G}$-orbit produce a finite product of actions on quasi-trees, and finally, the distance formula finishes the proof of Theorem 1.1.

The height of a finitely generated subgroup $H$ in a finitely generated group $G$ is the maximal $n\in\mathbb{N}$ such that there are distinct cosets $g_{1}H,\dots,g_{n}H\in G/H$ such that the intersection $g_{1}Hg_{1}^{-1}\cap\dots\cap g_{n}Hg_{n}^{-1}$ is infinite. The subgroup $H$ is called strongly quasi-convex in $G$ if for any $L\geq 1$, $C\geq 0$ there exists $R=R(L,C)$ such that every $(L,C)$–quasi-geodesic in $G$ with endpoints in $H$ is contained in the $R$–neighborhood of $H$. We note that strong quasiconvexity does not depend on the choice of finite generating set of the ambient group and it agrees with quasiconvexity when the ambient group is hyperbolic. In [GMRS98], the authors prove that quasi-convex subgroups in hyperbolic groups have finite height. It is a long-standing question asked by Swarup that whether or not the converse is true (see Question 1.8 in [Bes]). Tran in [Tra19] generalizes the result of [GMRS98] by showing that strongly quasi-convex subgroups in any finitely generated group have finite height. It is natural to ask whether or not the converse is true in this setting (i.e, finite height implies strong quasiconvexity). If the converse is true, then we could characterize strongly quasi-convex subgroup of a finitely generated group purely in terms of group theoretic notions.

In [NTY], the authors prove that having finite height and strong quasiconvexity are equivalent for all finitely generated $3$–manifold groups except the only ones containing the Sol command in its sphere-disk decomposition, and the graph manifold case was an essential case treated there. More precisely, Theorem 1.7 in [NTY] states that finitely generated subgroups of the fundamental group of a graph manifold are strongly quasi-convex if and only if they have finite height. The second main result of this paper is to generalize this result to Croke-Kleiner admissible action $G\curvearrowright X$.

We prove Theorem 1.6 by showing that $(\ref{thm1:item1})\Rightarrow(\ref{thm1:item2})\Rightarrow(\ref{thm1:item3})\Rightarrow(\ref{thm1:item1})$ and $(\ref{thm1:item3})\Leftrightarrow(\ref{thm1:item4})$. Similarly as in [NTY], the heart part of Theorem 1.6 is the implication $(\ref{thm1:item3})\Rightarrow(\ref{thm1:item1})$. We briefly review ideas in the proof of Theorem 1.7 in [NTY]. Suppose that $K$ is a finitely generated finite height subgroup of $\pi_{1}(M)$ where $M$ is a graph manifold. Let $M_{K}\to M$ be the covering space of $M$ corresponding to $K$. The authors in [NTY] prove that $K$ is strongly quasi-convex in $\pi_{1}(M)$ by using Sisto’s notion of path system $\mathcal{PS}(\tilde{M})$ in the universal cover $\tilde{M}$ of $M$, and prove that the preimage of the Scott core of $M_{K}$ in $\tilde{M}$ is $\mathcal{PS}(\tilde{M})$–contracting in the sense of Sisto. In this paper, the strategy of the proof of Theorem 1.6 is similar to the proof of Theorem 1.7 in [NTY] where we still use Sisto’s path system in $X$ but details are different. Sisto’s construction of special paths are carried out only in flip graph manifolds. Our construction of $(X,\mathcal{PS}(X))$ relies on the work of Croke-Kleiner [CK02] and applies to any admissible space $X$ (so any nonpostively curved graph manifold). We then construct a subspace $C_{K}\subset X$ on which $K$ acts geometrically and show that $C_{K}$ is contracting in $X$ with respect to the path system $(X,\mathcal{PS}(X))$. As a consequence, $K$ is strongly quasi-convex in $G$.

To conclude the introduction, we list a few questions and problems.

Quasi-isometric classification of graph manifolds has been studied by Kapovich-Leeb [KL98] and a complete quasi-isometric classification for fundamental groups of graph manifolds is given by Behrstock-Neumann [BN08]. Kapovich-Leeb prove that for any graph manifold $M$, there exists a flip graph manifold $N$ such that their fundamental groups are quasi-isometric. We would like to know that whether or not such a result holds for admissible groups.

With a positive answer to the above questions, we hope one can try to follow the strategy described in [BN08] to attack the following.

In [Liu13], Liu showed that the fundamental group of a non-positively curved graph manifold $M$ is virtually special (the case $\partial M\neq\varnothing$ was also obtained independently by Przytycki–Wise [PW14]). Thus, it is natural to ask the following.

As above, a positive answer to the question (with virtual compact specialness) would give an other proof of Theorem 1.1 under the same assumption.

In Section 2, we review some concepts and results about Croke-Kleiner admissible groups. In Section 3, we construct special paths in admissible spaces and give some results that will be used in the later sections. The proof of Theorem 1.5 and Proposition 1.4 is given in Section 4. We prove Theorem 1.1 and Theorem 1.6 in Section 5 and Section 6 respectively.

We would like to thank Chris Hruska, Hongbin Sun and Dani Wise for helpful conversations. W. Y. is supported by the National Natural Science Foundation of China (No. 11771022).

We use Lemma 3.3 in the proof of Proposition 3.8.

We are going to define a template associated to a geodesic in the Bass-Serre tree as the following.

The following lemma is cited from Lemma 4.5 and Proposition 4.6 in [CK02].

In this subsection, we are going to define special paths in $X$.

Recall that we choose a $G$–equivariant family of Euclidean planes $\{F_{e}:F_{e}\subset Y_{e}\}_{e\in T^{1}}$. For every pair of planes $(F_{e},F_{e^{\prime}})$ so that $v=e\cap e^{\prime}$, a minimal geodesic between $F_{e},F_{e^{\prime}}$ in $Y_{v}$ determines a strip $\mathcal{S}_{ee^{\prime}}=\gamma_{ee^{\prime}}\times\mathbb{R}$ for some geodesic $\gamma_{ee^{\prime}}\subset\overline{Y}_{v}$. It is possible that $\gamma_{ee^{\prime}}$ is trivial so the width of the strip is zero. Let $x\in X_{v}$ and $e$ an edge with an endpoint $v$. The minimal geodesic from $x$ to $F_{e}$ (possibly not belong to $Y_{v}$) also define a strip $\mathcal{S}_{xe}=\gamma_{xe}\times\mathbb{R}$ where the geodesic $\gamma_{xe}\subset\overline{Y}_{v}$ is the projection to $\overline{Y}_{v}$ of the intersection of this minimal geodesic with $Y_{v}$. Thus, $x$ is possibly not in the strip $\mathcal{S}_{xe}$ but within its $D$-neighborhood by Lemma 2.4.

In this section, we are going to prove the following proposition.

To get into the proof of Proposition 3.8, we need several lemmas (see Lemma 3.9 and Lemma 3.10).

Let $[x,y]$ be a geodesic in $Y_{v}=\overline{Y}_{v}\times\mathbb{R}$ with $y\in F_{e}$. We apply a minimizing horizontal slide of the endpoint $y\in F_{e}$ to obtain a point $z\in F_{e}$ so that $[y,z]$ is parallel to $\overline{Y}_{v}$ and the projection of $[x,z]$ on $\overline{Y}_{v}$ is orthogonal to $F_{e}\cap\overline{Y}_{v}$.

Let $G\curvearrowright X$ be a CKA action. Recall that each $Y_{v}$ decomposes as a metric product of a hyperbolic Hadamard space $\overline{Y}_{v}$ with the real line $\mathbb{R}$ such that $\overline{Y}_{v}$ admits a geometric action of $H_{v}$. Recall that we choose a $G$–equivariant family of Euclidean planes $\{F_{e}:F_{e}\subset Y_{e}\}_{e\in T^{1}}$.

Let $\mathbb{L}_{v}$ be the set of boundary lines of $\overline{Y}_{v}$ which are intersections of $\overline{Y}_{v}$ with $F_{e}$ for all edges $e$ issuing from $v$. Thus, there is a canonical one-to-one correspondence between $\mathbb{L}_{v}$ and the link of $v$ denoted by $Lk(v)$.

Construction of hyperbolic spaces $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$: We first partition the vertex set $T^{0}$ of the Bass-Serre tree into two disjoint class of vertices $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ such that if $v$ and $v^{\prime}$ are in $\mathcal{V}_{i}$ then $d_{T}(v,v^{\prime})$ is even.

Given $\mathcal{V}\in\{\mathcal{V}_{1},\mathcal{V}_{2}\}$, we shall build a geodesic (non-proper) hyperbolic space $\mathcal{X}$ by glueing $\overline{Y}_{v}$ for all $v\in\mathcal{V}$ along the boundary lines via flat links.

Consider the set of vertices in $\mathcal{V}$ such that their pairwise distance in $T$ equals 2. Equivalently, it is the union of the links of every vertex $w\in T^{0}-\mathcal{V}$. For any $v_{1}\neq v_{2}\in Lk(w)$, the edges $e_{1}=[v_{1},w]$ and $e_{2}=[v_{2},w]$ determine two corresponding boundary lines $\ell_{1}\in\mathbb{L}(v_{1})$ and $\ell_{2}\in\mathbb{L}(v_{2})$ which are the intersections of $\overline{Y}_{v_{i}}$ with $F_{e_{i}}$ for $i=1,2$ respectively. There exists a canonical identification between $\ell_{1}$ and $\ell_{2}$ so that their $\mathbb{R}$–coordinates equal in the metric product $Y_{w}=\overline{Y}_{w}\times\mathbb{R}$.

Note the link $Lk(w)$ determines a flat link $Fl(w)$ so that the flat strips are one-to-one correspondence with $Lk(w)$. In equivalent terms, it is a metric product $Lk(w)\times\mathbb{R}$, where $\mathbb{R}$ is parallel to the binding line.

For each $w\in T^{0}-\mathcal{V}$, the set of hyperbolic spaces $\overline{Y}_{v}$ where $v\in Lk(w)$ are glued to the flat links $Fl(w)$ along the boundary lines of flat strips and of hyperbolic spaces with the identification indicated above. Therefore, we obtain a metric space $\mathcal{X}$ from the union of $\{\overline{Y}_{v}:v\in\mathcal{V}\}$ and flat links $\{Fl(w),w\in T^{0}-\mathcal{V}\}$.