Isoperimetric Profiles and Regular Embeddings of locally compact groups

TL;DR

The paper extends the framework of $L^p$-measure subgroup couplings to locally compact unimodular groups and proves a dynamical criterion linking regular embeddings to $L^ fty$-measure subgroup couplings that are coarsely $m$-to-$1$. It then shows that the existence of $L^p$-measure subgroup couplings (and $ abla$-type, $\varphi$-integrable variants) enforces monotonicity of the $L^p$-isoperimetric profile under regular or coarse embeddings, with sublinear versions. A constructive passage from regular embeddings to measure subgroup couplings is developed using discretizations and cocycles, enabling the monotonicity theorems to hold in amenable unimodular locally compact groups. Consequently, the $L^p$-isoperimetric profile is monotone under regular and coarse embeddings in this setting, linking dynamical coupling data to large-scale geometric invariants.

Abstract

In this article we extend the notion of $L^p$-measure subgroups couplings, a quantitative asymmetric version of measure equivalence that was introduced by Delabie, Koivisto, Le Maître and Tessera for finitely generated groups, to the setting of locally compact compactly generated unimodular groups. As an example of these couplings; using ideas from Bader and Rosendal, we prove a "dynamical criteria" for the existence of regular embeddings between amenable locally compact compactly generated unimodular groups, namely the existence of an $L^\infty$-measure subgroup coupling that is coarsely $m$-to-$1$. We also prove that the existence of an $L^p$-measure subgroup that is coarsely $m$-to-$1$ implies the monotonicity of the $L^p$-isoperimetric profile, as well as sublinear version of this result. As a corollary we obtain that the $L^p$-isoperimetric profile is monotonous under regular embeddings, as well as coarse embeddings, between amenable unimodular locally compact compactly generated groups.

Theorems & Definitions (82)

• Proposition 1.1: Section $0.2.C'_2$ in Gromov
• Definition 1.2: Definition $0.5.E_1$ in Gromov
• Definition 1.3
• Proposition 1.4
• Definition 1.5
• Theorem 1.6
• Theorem A: Theorem \ref{['teoprincipal']}
• Theorem 1.7
• Theorem 1.8
• Theorem B: Theorem \ref{['teoLp']}