Invariance of non-vanishing of first $l^p$-cohomology under $L^q$-Measured Equivalence

TL;DR

This paper investigates the invariance of the non-vanishing of first $l^p$-cohomology under $L^q$-Measured Equivalence, revealing new invariance properties and applications to hyperbolic groups, conformal dimensions, and 3-manifold groups.

Contribution

It proves the invariance of non-vanishing first $l^p$-cohomology under $L^q$-Measured Equivalence for non-amenable groups and explores related applications.

Findings

Non-vanishing of first $l^p$-cohomology is invariant under $L^q$-Measured Equivalence for non-amenable groups.

Conformal dimension of hyperbolic Coxeter groups with certain boundary properties is invariant under $L^q$-Measured Equivalence.

Finitely generated free groups and surface groups are not $L^1$-Measured Equivalent.

Abstract

The first $l^p$-cohomology is an algebro-analytical object attached to a finitely generated discrete group and introduced by M. Gromov. It is well known that it is invariant under quasi-isometry. In this article, we prove that the non-vanishing of the first $l^p$-cohomology of a non-amenable group is invariant under $L^q$-Measured Equiavalence (an equivalence relation introduced by Gromov), where $q\geq p$. We also discuss many applications of this result. We prove that for hyperbolic (in the sense of Gromov) Coxeter groups with boundaries having Combinatorial Loewner Property, conformal dimension (of the canonical conformal gauge) of the Gromov boundary is invariant under $L^q$-Measured Equivalence for some large $q$. We prove that the finitely generated free groups and surface groups are not $L^1$-Measured Equivalent. We also give a lower bound of the critical exponent for the first $l^p$-cohomology of any lattice in $SO(n,1)$. Finally, we discuss $L^q$-Measured Equivalence between non-amenable 3-manifold groups corresponding to Thurston's three geometries $\mathbb{H}^3$, $\mathbb{H}^2\times\mathbb{R}$ and $\widetilde{SL_2(\mathbb{R})}$.