Finitely presented simple groups and measure equivalence

TL;DR

This paper constructs explicit examples of finitely presented, Kazhdan, simple groups that are not measure equivalent, using their actions on products of buildings and analyzing their $L^2$-Betti numbers.

Contribution

It provides explicit families of such groups and introduces methods to distinguish them via $L^2$-Betti number analysis.

Findings

Explicit infinite families of non-measure equivalent simple groups

Use of $L^2$-Betti numbers to differentiate groups

Groups act as lattices on products of buildings

Abstract

We exhibit explicit infinite families of finitely presented, Kazhdan, simple groups that are pairwise not measure equivalent. These groups are lattices acting on products of buildings. We obtain the result by studying vanishing and non-vanishing of their $L^2$-Betti numbers.