Fixed points of finite groups acting on generalised Thompson groups

We study centralisers of finite order automorphisms of the generalised Thompson groups F (n,a) and conjugacy classes of finite subgroups in finite extensions of F (n,a). In particular, we show that centralisers of finite automorphisms in F (n,a) are either of type FPa or not finitely generated. As an application we deduce the following result about the Bredon type of such finite extensions: any finite extension of F (n,a), where the elements of finite order act on F (n,a) via conjugation with piecewise-linear homeomorphisms, is of type Bredon F-a. In particular, finite extensions of F = F (2,a) are of type Bredon F (a) .