Automorphism Groups of Trees: Generalities and Prescribed Local Actions

TL;DR

This paper reviews automorphism groups of trees, discusses Tits' simplicity theorem, and introduces two constructions of tree groups based on local actions, highlighting their fundamental properties.

Contribution

It provides an expanded overview of automorphism groups of trees and presents two novel constructions of tree groups with prescribed local actions.

Findings

Analysis of automorphism groups of trees

Presentation of two constructions: universal groups and k-closures

Discussion of properties and applications of these groups

Abstract

This article is an expanded version of the talks given by the authors at the Arbeitsgemeinschaft "Totally Disconnected Groups", held at Oberwolfach in October 2014. We recall the basic theory of automorphisms of trees and Tits' simplicity theorem, and present two constructions of tree groups via local actions with their basic properties: the universal group associated to a finite permutation group by M. Burger and S. Mozes, and the $k$-closures of a given group by C. Banks, M. Elder and G. Willis.