Irreducible lattices, invariant means, and commensurating actions

TL;DR

This paper investigates the rigidity of lattices through the lens of invariant means and commensurating actions, focusing on Property FM which concerns actions with invariant means having finite orbits.

Contribution

It introduces new insights into the rigidity properties of lattices by analyzing invariant means and commensurating actions, especially regarding Property FM.

Findings

Property FM implies certain rigidity phenomena in lattices.

Actions with invariant means on discrete sets have finite orbits under studied conditions.

The work links invariant means to structural properties of lattices.

Abstract

We study rigidity properties of lattices in terms of invariant means and commensurating actions (or actions on CAT(0) cube complexes). We notably study Property FM for groups, namely that any action on a discrete set with an invariant mean has a finite orbit.