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

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.

The approximation property and exactness of locally compact groups

“…Recent researches (see e.g., [1], [2], [21], [27], [32], [33]) show that non-discrete locally compact groups also provide rich sources of interesting operator algebras. They also reveal attractive and fruitful interactions between locally compact group theory and theory of operator algebras.…”

Unitary representations of totally disconnected locally compact groups satisfying Ol'shanskii's factorization

“…In chapters 4, 5 and 6, we show that this framework applies to certain families of automorphism groups of trees and right-angled buildings whose representation theory are not yet treated in the literature such as automorphism groups of semiregular trees satisfying property IP k introduced in [BEW15] or such as universal groups of certain semiregular right-angled buildings introduced in [DMSS18]. Furthermore, in a forthcoming paper [Sem21], this axiomatic framework is used to obtain a complete classifiaction of the irreducible representations of groups classified by Radu in [Rad17] thereby contributing to the Type the I conjecture on trees originally introduced in [Neb99] and highlighted in [HR19].…”

An algebraic characterisation of ample type I groupoids

“…Roughly speaking groups enjoying one of these properties have a well-behaved unitary dual. Originally studied in the context of Lie groups and algebraic groups [3,4,7,9], other classes of non-discrete groups were considered more recently [5,10]. The question which discrete groups are CCR and type I was answered conclusively by Thoma [29], characterising them as the virtually abelian groups.…”