Fixed points of finite groups acting on generalised Thompson groups

Description/Abstract

We study centralisers of finite order automorphisms of the
generalised Thompson groups Fn, ∞ and conjugacy classes of finite sub-
groups in finite extensions of Fn, ∞. In particular we show that centralisers of finite automorphisms in Fn, ∞ are either of type FP∞ or not finitely generated.
As an application we deduce the following result about the Bredon type
of such finite extensions: any finite extension of Fn, ∞, where the elements of finite order act on Fn, ∞ via conjugation with piecewise-linear
homeomorphisms, is of type Bredon F∞. In particular finite extensions
of F = F2,∞ are of type Bredon F∞.