Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras

TL;DR

This paper characterizes when groups acting on trees are of type I, revealing many are not, and employs von Neumann algebra techniques to establish non-amenability of their group algebras, advancing understanding of their structure.

Contribution

It introduces a novel von Neumann algebra approach to classify type I groups acting on trees, including a complete classification for Burger-Mozes groups.

Findings

Many groups acting on trees are not of type I.

Group von Neumann algebras of these groups are non-amenable.

Complete classification of type I Burger-Mozes groups.

Abstract

We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown. We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger-Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.