Work

Measure equivalence rigidity among the Higman groups

Year: 2025

Type: article

Abstract: We prove that all (generalized) Higman groups on at least 5 generators are superrigid for measure equivalence. More precisely, let k\ge 5 , and let H be a group with generators a_{1},\dots,a_{k} and Baumslag–Solitar relations given by a_{i}a_{i+1}^{m_i}a_{i}^{-1}=a_{i}^{n_i} , with i varying in \mathbb{Z}/k\mathbb{Z} and non-zero integers |m_{i}|\neq |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 \mathrm{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.

Source: Journal of the European Mathematical Society

DOI: 10.4171/jems/1614

Authors: Camille Horbez, Jingyin Huang

Institutions: Centre National de la Recherche Scientifique, Université Paris-Saclay, Laboratoire de Mathématiques d'Orsay, The Ohio State University

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Agence Nationale de la Recherche

Awards: Artin groups, mapping class groups and Out(Fn): from geometry to operator algebras via measure equivalence, Dynamics of Automorphism of Groups: Growth, Entropy, Random walks