A connection between Lipschitz and Kazhdan constants for groups of homeomorphisms of the real line

Ignacio Vergara Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central 9170020, Chile [email protected]

Abstract.

We exhibit an obstruction for groups with Property (T) to act on the real line by bi-Lipschitz homeomorphisms. This condition is expressed in terms of the Lipschitz constants and the Kazhdan constants associated to finite, generating subsets. As a corollary, we obtain an upper bound for the Kazhdan constants of orderable groups. Our main tool is the Koopman representation associated to the action $\operatorname{BiLip}_{+}({\mathbb{R}})\curvearrowright{\mathbb{R}}$ .

Key words and phrases:

Property (T), Bi-Lipschitz homeomorphisms of the real line, orderable groups

2020 Mathematics Subject Classification:

Primary 22D55; Secondary 37E05, 22F10, 20F60

This work is supported by the FONDECYT project 3230024 and the ECOS project 23003 Small spaces under action

1. Introduction

We say that a finitely generated group $G$ has Property (T) if there is a finite subset $S\subset G$ and a constant $\varepsilon>0$ such that, for every unitary representation $\pi$ of $G$ on a Hilbert space $\mathcal{H}$ ,

\displaystyle\max_{g\in S}\|\pi(g)\xi-\xi\|\geq\varepsilon\|\xi\|,\quad\forall\xi\in(\mathcal{H}^{\pi})^{\perp},

(1)

where $(\mathcal{H}^{\pi})^{\perp}$ denotes the orthogonal complement of the subspace of $\pi$ -invariant vectors. We refer the reader to [2] for a detailed account on Property (T).

It is an open problem to determine whether a group $G$ with Property (T) can act faithfully on ${\mathbb{R}}$ by orientation-preserving homeomorphisms. The existence of such an action is equivalent to the existence of a left-invariant order on $G$ ; see [5, §1.1.3]. Furthermore, it was shown in [4, Theorem 8.5] that, in this case, there is always an action by bi-Lipschitz homeomorphisms with bounded displacement.

Let $\operatorname{Homeo}_{+}({\mathbb{R}})$ denote the group of orientation-preserving (i.e. increasing) homeomorphisms of ${\mathbb{R}}$ . We say that $f\in\operatorname{Homeo}_{+}({\mathbb{R}})$ is bi-Lipschitz if there is a constant $L\geq 1$ such that

\displaystyle\frac{1}{L}|x-y|\leq|f(x)-f(y)|\leq L|x-y|,\quad\forall x,y\in{\mathbb{R}}.

The Lipschitz constant $\operatorname{Lip}(f)$ is the infimum of all $L\geq 1$ such that the condition above holds.

Remark 1.1.

Let $f\in\operatorname{Homeo}_{+}({\mathbb{R}})$ be a bi-Lipschitz homoeomorphism, and let $Df$ denote its derivative function, which is defined almost everywhere. Then

\displaystyle\operatorname{Lip}(f)=\max\left\{\|Df\|_{\infty},\|Df^{-1}\|_{\infty}\right\},

where $\|\,\cdot\,\|_{\infty}$ stands for the essential supremum norm.

We say that $f$ has bounded displacement if

\displaystyle\sup_{x\in{\mathbb{R}}}|f(x)-x|<\infty.

We let $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ denote the subgroup of $\operatorname{Homeo}_{+}({\mathbb{R}})$ given by all bi-Lipschitz functions with bounded displacement. We leave as an exercise to the reader to check that $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ is indeed a group.

We are interested in finding obstructions for groups with Property (T) to be realised as subgroups of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ . The following result seems to be known to experts; we include a proof of it in Section 2 for the sake of completeness.

Proposition 1.2.

Let $G$ be a finitely generated group. Assume that there is a sequence of injective group homomorphisms $\theta_{n}:G\to\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ such that

\displaystyle\lim_{n\to\infty}\operatorname{Lip}(\theta_{n}(g))=1,

for all $g\in G$ . Then there is a nontrivial group homomorphism from $G$ to $({\mathbb{R}},+)$ . In particular, $G$ has an infinite abelian quotient.

Since Property (T) passes to quotients, a group satisfying the hypotheses of Proposition 1.2 cannot have Property (T); see [2, Theorem 1.3.4] and [2, Corollary 1.3.6]. In other words, if $G$ is a group with Property (T), generated by a finite subset $S$ , and $G$ acts on ${\mathbb{R}}$ by homeomorphisms in $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ , then

\displaystyle L=\max_{g\in S}\operatorname{Lip}(g)

cannot be arbitrarily small. The main goal of this paper is to obtain a quantitative version of this fact, which will be expressed in terms of the Kazhdan constant of $S$ .

Let $G$ be a group satisfying Property (T). This means that there is a pair $(S,\varepsilon)$ satisfying (1) for every unitary representation of $G$ . We call $(S,\varepsilon)$ a Kazhdan pair for $G$ . One can show that, if such a pair exists, then $S$ is necessarily a generating set for $G$ . Moreover, in this case, for every finite generating set $S^{\prime}\subset G$ , there is $\varepsilon^{\prime}>0$ such that $(S^{\prime},\varepsilon^{\prime})$ is a Kazhdan pair; see [2, Proposition 1.3.2].

For every finite subset $S\subset G$ , we define

\displaystyle\kappa(S)=\sup\left\{\varepsilon\geq 0\ \mid\ (S,\varepsilon)\text{ is a Kazhdan pair for }G\right\}.

(2)

We call $\kappa(S)$ the Kazhdan constant associated to $S$ . Observe that $G$ has Property (T) if and only if $\kappa(S)>0$ for some (equivalently, any) finite generating set $S$ . By looking at the left regular representation on $\ell^{2}(G)$ , which is given by

\displaystyle\lambda(g)\xi(h)=\xi(g^{-1}h),\quad\forall g,h\in G,\ \forall\xi\in\ell^{2}(G),

one sees that $\kappa(S)$ cannot be greater than $\sqrt{2}$ . Indeed, let $\delta_{e}$ denote the delta function at the identity element of $G$ . Then, for every $g\neq e$ ,

\displaystyle\|\lambda(g)\delta_{e}-\delta_{e}\|=\sqrt{2}.

We say that a subgroup $G$ of $\operatorname{Homeo}_{+}({\mathbb{R}})$ has a global fixed point if there exists $x_{0}\in{\mathbb{R}}$ such that

\displaystyle g(x_{0})=x_{0},\quad\forall g\in G.

Remark 1.3.

In this context, having a global fixed point is equivalent to the existence of a bounded orbit, which can be seen from the identity

\displaystyle f\left(\sup_{g\in G}g(x)\right)=\sup_{g\in G}f(g(x))=\sup_{g\in G}g(x),\quad\forall f\in G.

Now we can state our main result.

Theorem 1.4.

Let $G$ be a finitely generated subgroup of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ without global fixed points. If $G$ satisfies Property (T), then, for every finite generating set $S\subset G$ ,

\displaystyle\max_{g\in S}\operatorname{Lip}(g)\geq\Phi\left(\kappa(S)\right),

where $\Phi:[0,\sqrt{2})\to[1,\infty)$ is defined as

\displaystyle\Phi(t)=\max\left\{e^{2t},4(2-t^{2})^{-2}\right\},\quad\forall t\in[0,\sqrt{2}).

(3)

Remark 1.5.

The function $\Phi$ in Theorem 1.4 is strictly increasing and surjective. We include below the graphs of the two functions involved in its definition.

Figure 1. Graphs of the functions involved in the definition of

\Phi

Theorem 1.4 can be reinterpreted in the following way. Let $G=\langle S\rangle$ be a group with Property (T). Then $G$ cannot act on ${\mathbb{R}}$ by homeomorphisms in $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ without global fixed points and in such a way that

\displaystyle\operatorname{Lip}(g)<\Phi\left(\kappa(S)\right),

for all $g\in S$ .

The main ingredient in the proof of Theorem 1.4 is the Koopman representation on $L^{p}({\mathbb{R}})$ , associated to the action $G\curvearrowright{\mathbb{R}}$ , together with a version of Property (T) for $L^{p}$ spaces, as studied in [1]. The two functions involved in the definition of $\Phi$ are obtained from the cases $p=2$ and $p\to\infty$ .

As mentioned above, every orderable group can be realised as a subgroup of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ without global fixed points. We say that a group $G$ is (left) orderable if there is a total order $\prec$ on $G$ such that

\displaystyle g\prec f\quad\implies\quad hg\prec hf,

for all $f,g,h\in G$ . We refer the reader to [5] for a thorough account on orderable groups. The main motivation behind this note is the following open problem; see [5, Remark 3.5.21], and [8, Question 3] for a related question.

Question 1.6.

Does there exist an orderable group satisfying Property (T)?

We do not aim to answer this question here, but as a consequence of Theorem 1.4, we can obtain restrictions for the values of the Kazhdan constants of such groups, if they exist.

Corollary 1.7.

Let $G$ be a finitely generated, orderable group satisfying Property (T). Then, for every finite, symmetric generating subset $S\subset G$ ,

\displaystyle\kappa(S)\leq\Phi^{-1}(|S|),

where $\Phi$ is defined as in (3).

Observe that $\Phi^{-1}:[1,\infty)\to[0,\sqrt{2})$ is given by

\displaystyle\Phi^{-1}(t)=\min\left\{\tfrac{1}{2}\log(t),\sqrt{2}\left(1-t^{-1/2}\right)^{1/2}\right\},\quad\forall t\in[1,\infty).

For clarity, we include below the graphs of the two functions involved in its definition.

Figure 2. Graphs of the functions involved in the definition of

\Phi^{-1}

Organisation of the paper

Section 2 is devoted to the proof of Proposition 1.2. In Section 3, we prove some preliminary results regarding the Koopman representation on $L^{p}({\mathbb{R}})$ . Finally, in Section 4, we prove Theorem 1.4 and Corollary 1.7.

2. Limits of actions with arbitrarily small Lipschitz constants

In this section, we prove Proposition 1.2. The main idea is to construct an action by translations as a limit of actions with Lipschitz constants tending to $1$ . This will be achieved through the use of ultralimits. In order to do this, we need to consider actions with a uniform control on the displacement.

Lemma 2.1.

Let $g\in\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ . For $\alpha\in{\mathbb{R}}\setminus\{0\}$ , let $\varphi_{\alpha}:{\mathbb{R}}\to{\mathbb{R}}$ denote the multiplication by $\alpha$ . Then

\displaystyle\operatorname{Lip}(\varphi_{\alpha}^{-1}\circ g\circ\varphi_{\alpha})=\operatorname{Lip}(g),

and

\displaystyle\sup_{x\in{\mathbb{R}}}|\varphi_{\alpha}^{-1}\circ g\circ\varphi_{\alpha}(x)-x|=\alpha^{-1}\sup_{x\in{\mathbb{R}}}|g(x)-x|.

Proof.

For almost every $x\in{\mathbb{R}}$ ,

	$\displaystyle D(\varphi_{\alpha}^{-1}\circ g\circ\varphi_{\alpha})(x)$	$\displaystyle=Dg(\alpha x),$
	$\displaystyle D(\varphi_{\alpha}^{-1}\circ g^{-1}\circ\varphi_{\alpha})(x)$	$\displaystyle=Dg^{-1}(\alpha x).$

This shows that $\operatorname{Lip}(\varphi_{\alpha}^{-1}\circ g\circ\varphi_{\alpha})=\operatorname{Lip}(g)$ ; see Remark 1.1. On the other hand,

	$\displaystyle\sup_{x\in{\mathbb{R}}}\|\varphi_{\alpha}^{-1}\circ g\circ\varphi_{\alpha}(x)-x\|$	$\displaystyle=\sup_{x\in{\mathbb{R}}}\|\alpha^{-1}g(\alpha x)-\alpha^{-1}\alpha x\|$
		$\displaystyle=\alpha^{-1}\sup_{x\in{\mathbb{R}}}\|g(x)-x\|.$

∎

This result says that any $G<\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ can be conjugated into another subgroup of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ with the same Lipschitz constants, but for which the displacements are rescaled by a fixed amount.

Now we can prove Proposition 1.2. For details on ultrafilters and ultralimits, we refer the reader to [6]; see also [7].

Proof of Proposition 1.2.

Let $S\subset G$ be a finite generating subset. Conjugating the action $\theta_{n}$ by a homothecy, by Lemma 2.1, we may assume that

\displaystyle\max_{g\in S}\sup_{x\in{\mathbb{R}}}|\theta_{n}(g)(x)-x|=1,

for all $n\in\mathbb{N}$ . In particular, since $S$ generates $G$ , for all $x\in{\mathbb{R}}$ and $g\in G$ , the sequence $(\theta_{n}(g)(x))_{n\in\mathbb{N}}$ is bounded. Moreover, replacing $(\theta_{n})$ by a subsequence, we can find $g_{0}\in S$ such that

\displaystyle\sup_{x\in{\mathbb{R}}}|\theta_{n}(g_{0})(x)-x|=\max_{g\in S}\sup_{x\in{\mathbb{R}}}|\theta_{n}(g)(x)-x|=1,

(4)

for all $n\in\mathbb{N}$ . Fix a non-principal ultrafilter $\mathfrak{U}$ on $\mathbb{N}$ , and define

\displaystyle g\cdot x=\lim_{\mathfrak{U}}\theta_{n}(g)(x),\quad\forall g\in G,\ \forall x\in{\mathbb{R}}.

(5)

We claim that this defines an action by translations on ${\mathbb{R}}$ . Indeed, let $e$ denote the identity element of $G$ . Then

\displaystyle e\cdot x-x=\lim_{\mathfrak{U}}\theta_{n}(e)(x)-x=0,

for all $x\in{\mathbb{R}}$ . Moreover, for every $f,g\in G$ ,

	$\displaystyle(fg)\cdot x-f\cdot g\cdot x$	$\displaystyle=\lim_{\mathfrak{U}}\theta_{n}(fg)(x)-\theta_{n}(f)(g\cdot x)$
		$\displaystyle=\lim_{\mathfrak{U}}\theta_{n}(f)(\theta_{n}(g)(x)-g\cdot x).$

By hypothesis, there is a constant $C_{f}\geq 1$ such that $\operatorname{Lip}(\theta_{n}(f))\leq C_{f}$ for all $n\in\mathbb{N}$ . Thus

\displaystyle|(fg)\cdot x-f\cdot g\cdot x|\leq C_{f}\lim_{\mathfrak{U}}|\theta_{n}(g)(x)-g\cdot x|=0.

This shows that (5) defines an action of $G$ on ${\mathbb{R}}$ . Moreover, for all $g\in G$ and $x,y\in{\mathbb{R}}$ ,

\displaystyle|g\cdot x-g\cdot y|\leq\lim_{\mathfrak{U}}\operatorname{Lip}(\theta_{n}(g))|x-y|=|x-y|.

Applying the same reasoning to $g^{-1}$ , we obtain

\displaystyle|g\cdot x-g\cdot y|=|x-y|.

Therefore (5) defines an action by translations. Finally, the action is not trivial because

\displaystyle|g_{0}\cdot x-x|=\lim_{\mathfrak{U}}|\theta_{n}(g_{0})(x)-x|=1,

where $g_{0}$ is as in (4). The map $g\mapsto g\cdot 0$ is a group homomorphism from $G$ to ${\mathbb{R}}$ with infinite image. ∎

3. Property (T) and representations on $L^{p}$

3.1. Orthogonal representations

Let $G$ be a group, and let $E$ be a real Banach space. The orthogonal group $\mathbf{O}(E)$ is the group of linear isometries of $E$ . A group homomorphism $\pi:G\to\mathbf{O}(E)$ is called an orthogonal (or isometric) representation of $G$ on $E$ . We say that $\xi\in E$ is an invariant vector for the representation $\pi$ if

\displaystyle\pi(g)\xi=\xi,\quad\forall g\in G.

If $\mathcal{H}$ is a Hilbert space over $\mathbb{C}$ , the group of linear isometries of $\mathcal{H}$ is called the unitary group of $\mathcal{H}$ , and it is denoted by $\mathbf{U}(\mathcal{H})$ . A group homomorphism $\pi:G\to\mathbf{U}(\mathcal{H})$ is called a unitary representation of $G$ . As mentioned in the introduction, Property (T) is defined in terms of unitary representations; however, we will only deal with representations on real spaces here. The following fact is a consequence of the complexification of orthogonal representations; see [2, Remark A.7.2] and [2, Remark 2.12.1]. We record it here for completeness.

Lemma 3.1.

Let $G$ be a group with Property (T), and let $\pi:G\to\mathbf{O}(\mathcal{H})$ be an orthogonal representation on a real Hilbert space $\mathcal{H}$ . Assume that $\pi$ does not have nontrivial invariant vectors. Then, for every finite generating set $S\subset G$ ,

\displaystyle\max_{g\in S}\|\pi(g)\xi-\xi\|\geq\kappa(S)\|\xi\|,\quad\forall\xi\in\mathcal{H},

where $\kappa(S)$ is the Kazhdan constant associated to $S$ , as defined in (2).

3.2. The Mazur map and representations on $L^{p}$

In order to obtain the exponential bound in Theorem 1.4, we need to consider representations on $L^{p}({\mathbb{R}})$ for $p\geq 2$ . The Mazur map allows one to relate such representations for different values of $p$ . Let $(X,\mu)$ be a measure space, and let $1\leq p,q<\infty$ . The Mazur map $M_{q,p}:L^{q}(X,\mu)\to L^{p}(X,\mu)$ is defined by

\displaystyle M_{q,p}(\xi)=\operatorname{sign}(\xi)|\xi|^{\frac{q}{p}},\quad\forall\xi\in L^{q}(X,\mu).

(6)

The proof the following result can be found in [3, §9.1]; see also [9, §3.7.1].

Theorem 3.2 (Mazur).

Let $(X,\mu)$ be a measure space, and let $1\leq q\leq p<\infty$ . Then the Mazur map (6) is a homeomorphism between the unit spheres of $L^{q}(X,\mu)$ and $L^{p}(X,\mu)$ . More precisely, there is constant $C>0$ (depending only on $\frac{q}{p}$ ) such that

\displaystyle\frac{q}{p}\|\xi-\eta\|_{q}\leq\|M_{q,p}(\xi)-M_{q,p}(\eta)\|_{p}\leq C\|\xi-\eta\|_{q}^{\frac{q}{p}},

for all $\xi,\eta$ in the unit sphere of $L^{q}(X,\mu)$ .

Although the Mazur map is nonlinear, it conjugates orthogonal representations into orthogonal representations. This is a consequence of the Banach–Lamperti theorem, which describes the isometries of $L^{p}$ ; see e.g. [1, Theorem 2.16]. Moreover, it was shown in [1, Theorem A] that a group with Property (T) satisfies an analogous property for representations on $L^{p}$ spaces. We will need a quantitative version of this fact.

Proposition 3.3.

Let $(X,\mu)$ be a $\sigma$ -finite measure space and $p\in[2,\infty)$ . Let $G$ be a group with Property (T), and let $\pi:G\to\mathbf{O}(L^{p}(X,\mu))$ be an orthogonal representation without nontrivial invariant vectors. Then, for every finite generating subset $S\subset G$ ,

\displaystyle\max_{g\in S}\|\pi(g)\xi-\xi\|_{p}\geq\frac{2\kappa(S)}{p}\|\xi\|_{p},\quad\forall\xi\in L^{p}(X,\mu),

where $\kappa(S)$ is the Kazhdan constant associated to $S$ , as defined in (2).

Proof.

By [1, Lemma 4.2], we have an orthogonal representation $\tilde{\pi}:G\to\mathbf{O}(L^{2}(X,\mu))$ given by

\displaystyle\tilde{\pi}(g)=M_{p,2}\circ\pi(g)\circ M_{2,p},\quad\forall g\in G,

where $M_{2,p}$ and $M_{p,2}$ are defined as in (6). Moreover, by Theorem 3.2, $\tilde{\pi}$ does not have nontrivial invariant vectors. Now let $\xi$ be a unit vector in $L^{p}(X,\mu)$ and $g\in S$ . Again, by Theorem 3.2,

	$\displaystyle\\|\pi(g)\xi-\xi\\|_{p}$	$\displaystyle=\\|M_{2,p}\circ\tilde{\pi}(g)\circ M_{p,2}(\xi)-M_{2,p}\circ M_{p,2}(\xi)\\|_{p}$
		$\displaystyle\geq\frac{2}{p}\\|\tilde{\pi}(g)M_{p,2}(\xi)-M_{p,2}(\xi)\\|_{2}.$

Now, using the fact that $(S,\kappa(S))$ is a Kazhdan pair, we obtain

\displaystyle\max_{g\in S}\|\pi(g)\xi-\xi\|_{p}\geq\frac{2}{p}\kappa(S)\|M_{p,2}(\xi)\|_{2}=\frac{2\kappa(S)}{p}.

This holds for every unit vector in $L^{p}(X,\mu)$ . By homogeneity, we obtain the desired inequality for every $\xi\in L^{p}(X,\mu)$ . ∎

3.3. The Koopman representation for subgroups of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$

Let $G$ be a group acting on a measure space $(X,\mu)$ by measure class preserving transformations. This means that, for every $g\in G$ , the measures $\mu$ and $g_{*}\mu$ are absolutely continuous with respect to each other. Then, for every $p\in[1,\infty)$ , we can define the Koopman representation $\pi:G\to\mathbf{O}(L^{p}(X,\mu))$ by

\displaystyle\pi(g)\xi(x)=\xi(g^{-1}(x))\left(\frac{d(g_{*}\mu)}{d\mu}(x)\right)^{\frac{1}{p}},\quad\forall g\in G,\ \forall\xi\in L^{p}(X,\mu),\ \forall x\in X.

If $G$ is a subgroup of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ , then the action of $G$ on ${\mathbb{R}}$ preserves the measure class of the Lebesgue measure. In this case, the Koopman representation $\pi:G\to\mathbf{O}(L^{p}({\mathbb{R}}))$ is given by

\displaystyle\pi(g)\xi(x)=\xi(g^{-1}(x))Dg^{-1}(x)^{\frac{1}{p}},\quad\forall g\in G,\ \forall\xi\in L^{p}({\mathbb{R}}),\ \forall x\in{\mathbb{R}}.

(7)

This representation will be the main ingredient in the proof of Theorem 1.4. First, we need to establish that, in our setting, it does not have nontrivial invariant vectors.

Lemma 3.4.

Let $G$ be a finitely generated subgroup of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ , and let $\pi:G\to\mathbf{O}(L^{p}({\mathbb{R}}))$ be the associated Koopman representation for $p\in[1,\infty)$ . If $G$ does not have global fixed points, then $\pi$ does not have non-trivial invariant vectors.

Proof.

Assume by contradiction that $\xi$ is a $\pi$ -invariant vector with $\|\xi\|_{p}=1$ . Take $M>0$ such that

\displaystyle\int_{-M}^{M}|\xi(x)|^{p}\,dx>\frac{1}{2}.

By Remark 1.3, the action $G\curvearrowright{\mathbb{R}}$ has unbounded orbits. Hence there is $g\in G$ such that

\displaystyle[-M,M]\cap[g(-M),g(M)]=\varnothing.

Since $\pi(g)\xi=\xi$ , we have

	$\displaystyle 1$	$\displaystyle=\\|\xi\\|_{p}^{p}$
		$\displaystyle\geq\int_{-M}^{M}\|\xi(x)\|^{p}\,dx+\int_{g(-M)}^{g(M)}\|\pi(g)\xi(x)\|^{p}\,dx$
		$\displaystyle=2\int_{-M}^{M}\|\xi(x)\|^{p}\,dx$
		$\displaystyle>1,$

which is a contradiction. ∎

4. Proofs of the main results

In this section, we focus on Theorem 1.4 and Corollary 1.7. We will prove that, under the hypotheses of Theorem 1.4,

\displaystyle\kappa(S)\leq\Phi^{-1}\left(\max_{g\in S}\operatorname{Lip}(g)\right),

(8)

where $\Phi^{-1}:[1,\infty)\to[0,\sqrt{2})$ is given by

\displaystyle\Phi^{-1}(t)=\min\left\{\tfrac{1}{2}\log(t),\sqrt{2}\left(1-t^{-1/2}\right)^{1/2}\right\},\quad\forall t\in[1,\infty).

4.1. An upper bound for $\kappa(S)$ for large values of $\operatorname{Lip}(g)$

We begin with the bound $\kappa(S)\leq\sqrt{2}\left(1-\operatorname{Lip}(g)^{-1/2}\right)^{1/2}$ in (8).

Lemma 4.1.

\displaystyle\kappa(S)\leq\max_{g\in S}\sqrt{2}\left(1-\operatorname{Lip}(g)^{-1/2}\right)^{1/2}.

Proof.

Let $\pi:G\to\mathbf{O}(L^{2}({\mathbb{R}}))$ be the Koopman representation, defined as in (7). By Lemma 3.4, this representation does not have nontrivial invariant vectors. Fix a finite generating set $S\subset G$ , and define

	$\displaystyle M$	$\displaystyle=\max_{g\in S}\sup_{x\in{\mathbb{R}}}\|g(x)-x\|,$
	$\displaystyle L$	$\displaystyle=\max_{g\in S}\operatorname{Lip}(g).$

For every $n\geq 1$ , let us define a unit vector $\xi_{n}\in L^{2}({\mathbb{R}})$ by

\displaystyle\xi_{n}=(2n)^{-\frac{1}{2}}\mathds{1}_{[-n,n]},

where $\mathds{1}_{[-n,n]}$ denotes the indicator function of the interval $[-n,n]$ . For every $g\in S$ ,

\displaystyle\|\pi(g)\xi_{n}-\xi_{n}\|^{2}=2-2\langle\pi(g)\xi_{n},\xi_{n}\rangle,

and

	$\displaystyle\langle\pi(g)\xi_{n},\xi_{n}\rangle$	$\displaystyle=(2n)^{-1}\int_{-n}^{n}\mathds{1}_{[g(-n),g(n)]}(x)Dg^{-1}(x)^{\frac{1}{2}}\,dx$
		$\displaystyle\geq(2n)^{-1}L^{-\frac{1}{2}}\int_{-n}^{n}\mathds{1}_{[g(-n),g(n)]}(x)\,dx.$

For $n$ large enough, we have

\displaystyle\max\{-n,g(-n)\}\leq-n+M\leq n-M\leq\min\{n,g(n)\},

which shows that

	$\displaystyle\langle\pi(g)\xi_{n},\xi_{n}\rangle$	$\displaystyle\geq(2n)^{-1}L^{-\frac{1}{2}}(n-M-(-n+M))$
		$\displaystyle=\frac{n-M}{n}L^{-\frac{1}{2}}.$

Hence, since $\|\xi_{n}\|=1$ , by Lemma 3.1,

	$\displaystyle\kappa(S)^{2}$	$\displaystyle\leq\\|\pi(g)\xi_{n}-\xi_{n}\\|^{2}$
		$\displaystyle\leq 2-\frac{2(n-M)}{n}L^{-\frac{1}{2}}$

Taking the limit $n\to\infty$ , we obtain

	$\displaystyle\kappa(S)$	$\displaystyle\leq\sqrt{2}\left(1-L^{-\frac{1}{2}}\right)^{\frac{1}{2}}$
		$\displaystyle=\max_{g\in S}\sqrt{2}\left(1-\operatorname{Lip}(g)^{-1/2}\right)^{1/2}.$

∎

4.2. An upper bound for $\kappa(S)$ for small values of $\operatorname{Lip}(g)$

Now we deal with the logarithmic bound in (8). For this purpose, we need to consider the Koopman representation on $L^{p}({\mathbb{R}})$ . We will make use of the following estimate.

Lemma 4.2.

Let $L>1$ , $x\in[L^{-1},L]$ , and $p>\log(L)$ . Then

\displaystyle|x^{\frac{1}{p}}-1|\leq\frac{\log(L)}{p-\log(L)}.

Proof.

We will use the following classical inequalities for the exponential function:

\displaystyle 1+t\leq e^{t}\leq 1+\frac{t}{1-t},\quad\forall t\in(-\infty,1).

For $t=\frac{1}{p}\log(x)$ , we get

\displaystyle\frac{1}{p}\log(x)\leq x^{\frac{1}{p}}-1\leq\frac{\log(x)}{p-\log(x)}.

Assume first that $x\geq 1$ . Then

\displaystyle|x^{\frac{1}{p}}-1|=x^{\frac{1}{p}}-1\leq\frac{\log(x)}{p-\log(x)}\leq\frac{\log(L)}{p-\log(L)}.

On the other hand, if $x<1$ , then

\displaystyle|x^{\frac{1}{p}}-1|=1-x^{\frac{1}{p}}\leq\frac{1}{p}\log(x^{-1})\leq\frac{\log(L)}{p}\leq\frac{\log(L)}{p-\log(L)}.

∎

Now we are ready to obtain our second estimate for $\kappa(S)$ .

Lemma 4.3.

\displaystyle\kappa(S)\leq\frac{1}{2}\max_{g\in S}\log(\operatorname{Lip}(g)).

Proof.

Let $S\subset G$ be a finite generating set. As in the proof of Lemma 4.1, let us define the following constants:

	$\displaystyle M$	$\displaystyle=\max_{g\in S}\sup_{x\in{\mathbb{R}}}\|g(x)-x\|,$
	$\displaystyle L$	$\displaystyle=\max_{g\in S}\operatorname{Lip}(g).$

Let $p>\log(L)$ , and let $\pi:G\to\mathbf{O}(L^{p}({\mathbb{R}}))$ be the Koopman representation:

\displaystyle\pi(g)\xi(x)=\xi(g^{-1}(x))Dg^{-1}(x)^{\frac{1}{p}},\quad\forall g\in G,\ \forall\xi\in L^{p}({\mathbb{R}}),\ \forall x\in{\mathbb{R}}.

By Lemma 3.4, $\pi$ does not have nontrivial invariant vectors. Hence, by Proposition 3.3, we have

\displaystyle\max_{g\in S}\|\pi(g)\xi-\xi\|_{p}\geq\frac{2\kappa(S)}{p}\|\xi\|_{p},\quad\forall\xi\in L^{p}({\mathbb{R}}).

(9)

For every $n\geq 1$ , let us define

\displaystyle\xi_{n}=(2n)^{-\frac{1}{p}}\mathds{1}_{[-n,n]}.

Observe that $\|\xi_{n}\|_{p}=1$ for all $n\geq 1$ . On the other hand, for every $g\in S$ ,

\displaystyle\|\pi(g)\xi_{n}-\xi_{n}\|_{p}^{p}=(2n)^{-1}\int_{-\infty}^{\infty}\left|\mathds{1}_{[g(-n),g(n)]}(x)Dg^{-1}(x)^{\frac{1}{p}}-\mathds{1}_{[-n,n]}(x)\right|^{p}\,dx.

Now, for every $n$ large enough,

\displaystyle-n-M\leq\min\{-n,g(-n)\}\leq\max\{-n,g(-n)\}\leq-n+M,

and

\displaystyle n-M\leq\min\{n,g(n)\}\leq\max\{n,g(n)\}\leq n+M.

Hence

	$\displaystyle 2n\\|\pi(g)\xi_{n}-\xi_{n}\\|_{p}^{p}$	$\displaystyle\leq\max\left\{\int_{-n-M}^{-n+M}Dg^{-1}(x)\,dx,\int_{-n-M}^{-n+M}1\,dx\right\}$
		$\displaystyle\quad+\int_{-n-M}^{n+M}\left\|Dg^{-1}(x)^{\frac{1}{p}}-1\right\|^{p}\,dx$
		$\displaystyle\quad+\max\left\{\int_{n-M}^{n+M}Dg^{-1}(x)\,dx,\int_{n-M}^{n+M}1\,dx\right\}$
		$\displaystyle\leq 4ML+\int_{-n-M}^{n+M}\left\|Dg^{-1}(x)^{\frac{1}{p}}-1\right\|^{p}\,dx.$

By Lemma 4.2, for every $g\in S$ and every $n$ large enough,

\displaystyle\|\pi(g)\xi_{n}-\xi_{n}\|_{p}^{p}\leq\frac{4ML}{2n}+\frac{2(n+M)}{2n}\left(\frac{\log(L)}{p-\log(L)}\right)^{p}.

Therefore

\displaystyle\limsup_{n\to\infty}\max_{g\in S}\|\pi(g)\xi_{n}-\xi_{n}\|_{p}\leq\frac{\log(L)}{p-\log(L)}.

Combining this with (9), we get

\displaystyle\frac{2\kappa(S)}{p}\leq\frac{\log(L)}{p-\log(L)}.

Equivalently,

\displaystyle\kappa(S)\leq\frac{\log(L)}{2\left(1-\frac{1}{p}\log(L)\right)}.

Letting $p\to\infty$ , this yields

	$\displaystyle\kappa(S)$	$\displaystyle\leq\frac{1}{2}\log(L)$
		$\displaystyle=\frac{1}{2}\log\left(\max_{g\in S}\operatorname{Lip}(g)\right).$

∎

We can now prove Theorem 1.4.

Proof of Theorem 1.4.

Let $S$ be a finite generating set of $G$ , and let $\kappa(S)$ be its Kazhdan constant. By Lemmas 4.1 and 4.3,

\displaystyle\kappa(S)\leq\min\left\{\tfrac{1}{2}\log(L),\sqrt{2}\left(1-L^{-1/2}\right)^{1/2}\right\}=\Phi^{-1}(L),

where

\displaystyle L=\max_{g\in S}\operatorname{Lip}(g).

Hence $L\geq\Phi(\kappa(S))$ , where $\Phi$ is defined as in (3). ∎

4.3. Consequences for orderable groups

Now we prove Corollary 1.7 as a consequence of Theorem 1.4. The main ingredient in the proof is the following result, which was essentially proved in [4, Theorem 8.5].

Theorem 4.4 (Deroin–Kleptsyn–Navas–Parwani).

Let $G$ be a finitely generated, orderable group. Let $S$ be a finite, symmetric generating subset of $G$ . Then $G$ is isomorphic to a subgroup of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ without global fixed points. Moreover, for every $g\in S$ ,

\displaystyle\operatorname{Lip}(g)\leq|S|.

(10)

Proof.

By [5, Proposition 1.1.8], we can identify $G$ with a subgroup of $\operatorname{Homeo}_{+}({\mathbb{R}})$ without global fixed points. By [4, Theorem 8.5], it can be conjugated to a subgroup of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ . Finally, the estimate (10) follows from (the proof of) [4, Proposition 8.4] applied to the probability measure

\displaystyle\mu(g)=\begin{cases}\frac{1}{|S|}&\text{if }g\in S,\\ 0&\text{otherwise}.\end{cases}

∎

Proof of Corollary 1.7.

Let $S$ be a finite, symmetric generating subset of $G$ . By Theorem 4.4, $G$ is isomorphic to a subgroup of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$ without global fixed points and such that $\operatorname{Lip}(g)\leq|S|$ for every $g\in S$ . Hence, by Theorem 1.4,

\displaystyle\kappa(S)\leq\Phi^{-1}(|S|).

∎

Acknowledgements

I am grateful to Andrés Navas for many interesting discussions, and for his very valuable comments and suggestions.

References

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[2] Bachir Bekka, Pierre de la Harpe, and Alain Valette. Kazhdan’s property (T), volume 11 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008.
[3] Yoav Benyamini and Joram Lindenstrauss. Geometric nonlinear functional analysis. Vol. 1, volume 48 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2000.
[4] Bertrand Deroin, Victor Kleptsyn, Andrés Navas, and Kamlesh Parwani. Symmetric random walks on ${\rm Homeo}^{+}({\bf R})$ . Ann. Probab., 41(3B):2066–2089, 2013.
[5] Bertrand Deroin, Andrés Navas, and Cristóbal Rivas. Groups, orders, and dynamics. arXiv preprint arXiv:1408.5805, 2014.
[6] Isaac Goldbring. Ultrafilters throughout mathematics, volume 220 of Grad. Stud. Math. American Mathematical Society (AMS), Providence, RI, 2022.
[7] Stefan Heinrich. Ultraproducts in Banach space theory. J. Reine Angew. Math., 313:72–104, 1980.
[8] Andrés Navas. Group actions on 1-manifolds: a list of very concrete open questions. In Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, pages 2035–2062. World Sci. Publ., Hackensack, NJ, 2018.
[9] Piotr W. Nowak. Group actions on Banach spaces. In Handbook of group actions. Vol. II, volume 32 of Adv. Lect. Math. (ALM), pages 121–149. Int. Press, Somerville, MA, 2015.

	$\displaystyle 1$	$\displaystyle=\\|\xi\\|_{p}^{p}$
		$\displaystyle\geq\int_{-M}^{M}\|\xi(x)\|^{p}\,dx+\int_{g(-M)}^{g(M)}\|\pi(g)\xi(x)\|^{p}\,dx$
		$\displaystyle=2\int_{-M}^{M}\|\xi(x)\|^{p}\,dx$
		$\displaystyle>1,$

	$\displaystyle 2n\\|\pi(g)\xi_{n}-\xi_{n}\\|_{p}^{p}$	$\displaystyle\leq\max\left\{\int_{-n-M}^{-n+M}Dg^{-1}(x)\,dx,\int_{-n-M}^{-n+M}1\,dx\right\}$
		$\displaystyle\quad+\int_{-n-M}^{n+M}\left\|Dg^{-1}(x)^{\frac{1}{p}}-1\right\|^{p}\,dx$
		$\displaystyle\quad+\max\left\{\int_{n-M}^{n+M}Dg^{-1}(x)\,dx,\int_{n-M}^{n+M}1\,dx\right\}$
		$\displaystyle\leq 4ML+\int_{-n-M}^{n+M}\left\|Dg^{-1}(x)^{\frac{1}{p}}-1\right\|^{p}\,dx.$

A connection between Lipschitz and Kazhdan constants for groups of homeomorphisms of the real line

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Remark 1.1.

Proposition 1.2.

Remark 1.3.

Theorem 1.4.

Remark 1.5.

Question 1.6.

Corollary 1.7.

Organisation of the paper

2. Limits of actions with arbitrarily small Lipschitz constants

Lemma 2.1.

Proof.

Proof of Proposition 1.2.

3. Property (T) and representations on LpL^{p}

3.1. Orthogonal representations

Lemma 3.1.

3.2. The Mazur map and representations on LpL^{p}

Theorem 3.2 (Mazur).

Proposition 3.3.

Proof.

3.3. The Koopman representation for subgroups of BiLip+bd⁡(ℝ)\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})

Lemma 3.4.

Proof.

4. Proofs of the main results

4.1. An upper bound for κ​(S)\kappa(S) for large values of Lip⁡(g)\operatorname{Lip}(g)

Lemma 4.1.

Proof.

4.2. An upper bound for κ​(S)\kappa(S) for small values of Lip⁡(g)\operatorname{Lip}(g)

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Proof of Theorem 1.4.

4.3. Consequences for orderable groups

Theorem 4.4 (Deroin–Kleptsyn–Navas–Parwani).

Proof.

Proof of Corollary 1.7.

Acknowledgements

References

3. Property (T) and representations on $L^{p}$

3.2. The Mazur map and representations on $L^{p}$

3.3. The Koopman representation for subgroups of $\operatorname{BiLip}_{+}^{\mathrm{bd}}({\mathbb{R}})$

4.1. An upper bound for $\kappa(S)$ for large values of $\operatorname{Lip}(g)$

4.2. An upper bound for $\kappa(S)$ for small values of $\operatorname{Lip}(g)$