Fixed points of finite groups acting on generalised Thompson groups

TL;DR

This paper investigates the structure of centralisers of finite order automorphisms and finite subgroups in generalized Thompson groups, revealing their finiteness properties and implications for Bredon type classifications.

Contribution

It provides new insights into the finiteness properties of centralisers and finite subgroups in generalized Thompson groups, extending understanding of their automorphism and extension structures.

Findings

Centralisers either admit finite type classifying spaces or are not finitely generated.

Results on the Bredon type of finite extensions of these groups.

Analysis of conjugacy classes of finite subgroups.

Abstract

We study centralisers of finite order automorphisms of generalisations of Thompson's group F and conjugacy classes of finite subgroups in finite extensions of these groups. In particular, we show that centralisers of finite automorphisms in these generalised Thompson groups either admit a finite type classifying space or are not finitely generated. As an application we deduce results about the Bredon type of such finite extensions.