Charmenability and Stiffness of Arithmetic Groups

TL;DR

This paper characterizes charmenability in arithmetic groups, establishing a dichotomy for their dynamics and representations, and proves stiffness of stationary measures, generalizing classical results and extending to higher rank cases.

Contribution

It introduces a new characterization of charmenability in arithmetic groups and proves stiffness of stationary measures in this context, extending classical results to higher rank groups.

Findings

Characterization of charmenability in arithmetic groups

Proof of stiffness for stationary measures in these groups

Extension of classical Furstenberg results to higher rank cases

Abstract

We characterize charmenability among arithmetic groups and deduce dichotomy statements pertaining normal subgroups, characters, dynamics, representations and associated operator algebras. We do this by studying the stationary dynamics on the space of characters of the amenable radical, and in particular we establish stiffness: any stationary probability measure is invariant. This generalizes a classical result of Furstenberg for dynamics on the torus. Under a higher rank assumption, we show that any action on the space of characters of a finitely generated virtually nilpotent group is stiff.