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On the stability of hyperbolicity under quantitative measure equivalence

Thiebout Delabie, Juhani Koivisto, François Le Maître, Romain Tessera

TL;DR

The paper investigates the stability of hyperbolicity under quantitative measure equivalence between finitely generated groups. It proves that a cobounded $(\mathrm{L}^p,\mathrm{L}^{\infty})$ measure equivalence coupling from a hyperbolic group $\Gamma$ to a group $\Lambda$ enforces hyperbolicity of $\Lambda$ for a threshold $p>108\delta\mathrm{Ent}(S_\Gamma)+2$, with the bound depending on the hyperbolicity constant $\delta$ and the volume entropy $\mathrm{Ent}(S_\Gamma)$. It extends to more general $(\varphi,\psi)$-integrable couplings, showing rigidity under exponential-polynomial trade-offs and highlighting a sharp contrast with higher-rank lattices where exponentials dominate. The argument combines a geometric obstruction based on embedded cycles in non-hyperbolic spaces with cocycle estimates arising from the ME coupling, and leverages coboundedness to relate fundamental domains. Overall, the work demonstrates a strong form of hyperbolicity rigidity under quantitative orbit/measure equivalence and yields concrete corollaries for rank-one lattices and OE settings.

Abstract

A well-known result of Shalom says that lattices in SO$(n,1)$ are $\mathrm{L}^p$ measure equivalent for all $p<n-1$. His proof actually yields the following stronger statement: the natural coupling resulting from a suitable choice of fundamental domains from a uniform lattice to a non-uniform one is $(\mathrm{L}^p,\mathrm{L}^{\infty})$. Moreover, it is easy to see that the coupling is cobounded: the fundamental domain of the uniform lattice is contained in a union of finitely many translates of the fundamental domain of the non-uniform one. The purpose of this note is to prove that this statement is sharp in the following sense: if a ME-coupling from a hyperbolic group to a non-hyperbolic group is cobounded and $(\mathrm{L}^p,\mathrm{L}^{\infty})$, then $p$ must be less than some $p_0$ only depending on the hyperbolic group.

On the stability of hyperbolicity under quantitative measure equivalence

TL;DR

The paper investigates the stability of hyperbolicity under quantitative measure equivalence between finitely generated groups. It proves that a cobounded measure equivalence coupling from a hyperbolic group to a group enforces hyperbolicity of for a threshold , with the bound depending on the hyperbolicity constant and the volume entropy . It extends to more general -integrable couplings, showing rigidity under exponential-polynomial trade-offs and highlighting a sharp contrast with higher-rank lattices where exponentials dominate. The argument combines a geometric obstruction based on embedded cycles in non-hyperbolic spaces with cocycle estimates arising from the ME coupling, and leverages coboundedness to relate fundamental domains. Overall, the work demonstrates a strong form of hyperbolicity rigidity under quantitative orbit/measure equivalence and yields concrete corollaries for rank-one lattices and OE settings.

Abstract

A well-known result of Shalom says that lattices in SO are measure equivalent for all . His proof actually yields the following stronger statement: the natural coupling resulting from a suitable choice of fundamental domains from a uniform lattice to a non-uniform one is . Moreover, it is easy to see that the coupling is cobounded: the fundamental domain of the uniform lattice is contained in a union of finitely many translates of the fundamental domain of the non-uniform one. The purpose of this note is to prove that this statement is sharp in the following sense: if a ME-coupling from a hyperbolic group to a non-hyperbolic group is cobounded and , then must be less than some only depending on the hyperbolic group.

Paper Structure

This paper contains 14 sections, 17 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Uniform versus non-uniform lattices in rank one simple Lie groups.
  3. Stability of hyperbolicity under quantitative orbit equivalence
  4. Hyperbolicity and embedded cycles.
  5. Sketch of proof.
  6. Further remarks and questions.
  7. Plan of the paper.
  8. Geometric preliminaries
  9. Preliminaries on quantitative measure equivalence
  10. Relations between fundamental domains
  11. phi,psi-integrable measure equivalence
  12. Rigidity of hyperbolicity
  13. Upper estimates for the restriction of $b_x$ to $C_n$.
  14. Lower estimates for the restriction of $b_x$ to $C_n$.

Key Result

Theorem 1

Let $\Gamma$ and $\Lambda$ be two lattices in SO$(n,1)$ such that $\Gamma$ is uniform. Then there exists a cobounded measure equivalence coupling from $\Gamma$ to $\Lambda$ that is $(\mathrm{L}^p,\mathrm{L}^{\infty})$-integrable for all $p<n-1$.

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem : Shalom shalomRigidityUnitaryRepresentations2000
  • Theorem 1: see Cor. \ref{['cor:Lp-infty']}
  • Remark 1.2
  • Corollary 2
  • Theorem : de la Salle
  • Theorem 3: see Cor. \ref{['cor: phi coupling hyp']}
  • Remark 1.4
  • Corollary 4
  • Remark 1.5
  • ...and 24 more