Measure equivalence rigidity among the Higman groups

2022/06/02 by Camille Horbez, Horbez, Camille, JingYin Huang +1 · 1 citation
Mathematics · Computer Science · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2206.00884

Abstract

We prove that all (generalized) Higman groups on at least 5 generators are superrigid for measure equivalence. More precisely, let k≥ 5, and let H be a group with generators a₁,…,a_k, and Baumslag-Solitar relations given by a_ia_i+1^m_ia_i^-1=a_i^n_i, with i varying in ℤ/kℤ and nonzero integers |m_i|≠ |n_i| for each i. We prove that every countable group which is measure equivalent to H, is in fact virtually isomorphic to H. A key ingredient in the proof is a general statement providing measured group theoretic invariants for groups acting acylindrically on CAT(-1) polyhedral complexes with control on vertex and edge stabilizers. Among consequences of our work, we obtain rigidity theorems for generalized Higman groups with respect to lattice embeddings and automorphisms of their Cayley graphs. We also derive an orbit equivalence and W^*-superrigidity theorem for all free, ergodic, probability measure-preserving actions of generalized Higman groups.

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