Towards computing the rational homology and assembly maps of generalised Thompson groups

TL;DR

This paper studies the algebraic and geometric properties of generalized Thompson groups, constructing specific complexes on which they act, and applies these findings to the Farrell-Jones assembly map in algebraic K-theory.

Contribution

It introduces a method to construct high-dimensional complexes with group actions for generalized Thompson groups and computes conjugacy classes of finite cyclic subgroups, advancing understanding of their algebraic K-theory.

Findings

Existence of k-dimensional, n-connected complexes with group actions

Explicit counts of conjugacy classes of finite cyclic subgroups

Application to rationalized Farrell-Jones assembly map

Abstract

Let $V_r(\Sigma)$ be the generalised Thompson group defined as the automorphism group of a valid, bounded, and complete Cantor algebra. We show that that for every $n>0$ there is a $k>n,$ such that there exists a $k$-dimensional $n$-connected simplicial complex $K$ such that $V_r(\Sigma)$ acts on $K$ with finite stabilisers. We also determine the number of conjugacy classes of finite cyclic subgroups of a given order $m$ in Brin-Thompson groups. We apply our computations to the rationalised Farrell-Jones assembly map in algebraic $K$-theory.