Directions and scale-multiplicative semigroups in restricted Burger-Mozes groups

Prime Localizations of Burger--Mozes-type groups

This article concerns Burger--Mozes universal groups acting on regular trees locally like a given permutation group of finite degree. We also consider locally isomorphic generalizations of the former due to Le Boudec and Lederle. For a large class of such permutation groups and primes $p$ we determine their local $p$-Sylow subgroups as well as subgroups of their $p$-localization, which is identified as a group of the same type in certain cases.

The universal group of Burger--Mozes and the Howe--Moore property

By constructing a new unitary representation we prove the universal group $U(F)^+$ of Burger--Mozes does not have the Howe--Moore property when $F$ is primitive but not $2$-transitive. It is well known $U(F)^+$ does have this property when $F$ is $2$-transitive. Along the way, we give a characterization of the universal group, when $F$ is primitive, to have the Howe--Moore property, and also prove $U(F)^+$ has the relative Howe--Moore property. These two results are a consequence of a strengthening of Mautner's phenomenon for locally compact groups acting on d-regular trees and having Tits' in…

Groups Acting on Trees With Prescribed Local Action

We extend Burger--Mozes theory of closed, non-discrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger--Mozes universal groups acting on the regular tree $T_{d}$ of degree $d\in\mathbb{N}_{\ge 3}$. Three applications are given: First, we characterize the automorphism types which the quasi-center of a non-discrete subgroup of $\mathrm{Aut}(T_{d})$ may feature in terms of the group's local~action. In doing so, we explicitly construct closed, non-discrete, compactly generated subgroups of $\mathrm{Au…

在Davis复形积中的简单晶格

Simple Lattices in Products of Davis Complexes

本文研究了在特定Davis复形积中构造简单一致晶格的方法，引入了Burger-Mozes通用群的类比，并提供了一个局部标准来判断顶点传递群在通用群中的稠密性。

Coloured Neretin Groups

We give sufficient conditions for a subgroup of a tree almost automorphism group to be isomorphic to the topological full groups of a one-sided shift in the sense of Matui. As an application, we show that almost automorphism groups of trees obtained from universal groups constructed by Burger and Mozes are compactly generated and virtually simple. In addition, using the approach of Bader, Caprace, Gelander and Mozes we show that some of these almost automorphism groups do not have any lattice.

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

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.