Charmenability of arithmetic groups of product type

TL;DR

This paper introduces the concepts of charmenability and charfiniteness for arithmetic groups of product type, exploring their properties and implications in dynamics, ergodic theory, and representation theory.

Contribution

It formalizes new properties of arithmetic groups related to their character spaces and positive definite functions, with applications to various areas of group theory and dynamics.

Findings

Defined charmenability and charfiniteness for these groups

Analyzed their implications in topological dynamics and ergodic theory

Applied concepts to groups acting on products of trees

Abstract

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees.