Radu groups acting on trees are CCR

TL;DR

This paper classifies certain unitary representations of automorphism groups of trees with specific symmetry properties, confirming a conjecture about their CCR nature for a broad class of semi-regular trees.

Contribution

It provides a classification of irreducible unitary representations for a class of automorphism groups of trees and verifies the CCR conjecture in new cases.

Findings

Classification of irreducible unitary representations of these groups

Confirmation of CCR conjecture for semi-regular trees with degrees in a dense set

Extension of known results to broader classes of trees

Abstract

We classify the irreducible unitary representations of closed simple groups of automorphisms of trees acting $2$-transitively on the boundary and whose local action at every vertex contains the alternating group. As an application, we confirm Claudio Nebbia's CCR conjecture on trees for $(d_0,d_1)$-semi-regular trees such that $d_0,d_1\in \Theta$, where $\Theta$ is an asymptotically dense set of positive integers.