A continuum of non-measure equivalent groups

TL;DR

This paper constructs a large family of countable groups with property (T), zero l^2-Betti numbers, and torsion-freeness, all mutually non-measure equivalent, expanding understanding of group measure equivalence classes.

Contribution

It introduces a continuum of non-measure equivalent groups with property (T), zero l^2-Betti numbers, and torsion-freeness, demonstrating the diversity within these properties.

Findings

Constructed a continuum of non-measure equivalent groups.

All groups have property (T), zero l^2-Betti numbers, and are torsion-free.

Groups are pairwise non-measure equivalent.

Abstract

We construct a continuum sized family $\{G_x\}_{x\in\{0,1\}^{\mathbb N}}$ of pairwise non-measure equivalent countable groups which have property (T) (hence are finitely generated), have zero $\ell^2$-Betti numbers of all orders, and are torsion-free.