On the stability of hyperbolicity under quantitative measure equivalence

TL;DR

This paper investigates the stability of hyperbolicity under measure equivalence, showing that certain couplings preserve hyperbolic properties only within specific measure bounds, highlighting the sharpness of previous results.

Contribution

It proves that cobounded, $(L^p,L^{ olinebreak}\infty)$ measure equivalence couplings from hyperbolic to non-hyperbolic groups are limited to $p$ below a certain threshold, confirming the optimality of earlier findings.

Findings

Couplings are $(L^p,L^{ olinebreak}\infty)$ only for $p < p_0$

The coupling's coboundedness is crucial for hyperbolicity stability

The result sharpens previous measure equivalence theorems

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.