Measure equivalence rigidity among the Higman groups

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 $|m_i|\ne |n_i|$ 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 $W^{\ast }$-superrigidity theorem for all free, ergodic, probability measure-preserving actions of generalized Higman groups.