Measure equivalence and sofic approximations

2025/06/26 by Thiebout Delabie, Delabie, Thiebout, Juhani Koivisto +3
Mathematics · #Advanced Operator Algebra Research #Mathematical Dynamics and Fractals #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2506.21750

Abstract

We introduce a technique for producing a measure coupling between two sofic groups from a family of maps between their sofic approximations. We exploit this to construct measure couplings between pairs of groups with prescribed integrability conditions. As an application we show that solvable Baumslag-Solitar groups, Lamplighters and the group SOL are all exponentially measure equivalent to one another: in particular they are L^p measure equivalent for all p. This is in sharp contrast with the fact that these groups are in general not quasi-isometric to one another: indeed, for instance the lamplighter with lamp group Z/3Z is not quasi-isometric the lamplighter with lamp group Z/2Z.

Citations

• L^p measure equivalence of nilpotent groups
• On the absence of quantitatively critical measure equivalence couplings
• Sofic approximations and quantitative measure couplings
• Quantitative measure equivalence between amenable groups
• Invariant measures for actions of congruent monotileable amenable groups
• Speed of random walks, isoperimetry and compression of finitely generated groups
• Ultraproducts, weak equivalence and sofic entropy
• Coarse differentiation of quasi-isometries II: Rigidity for Sol and\n Lamplighter groups
• Homological Invariants and Quasi-Isometry

Related

• Sofic approximations and quantitative measure couplings
• Quantitative measure equivalence between amenable groups
• Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity
• Sofic boundaries of groups and coarse geometry of sofic approximations
• On the absence of quantitatively critical measure equivalence couplings