The homology of the Higman-Thompson groups

TL;DR

This paper proves that Thompson's group V is acyclic by identifying its homology with that of a certain infinite loop space, using homological stability and algebraic K-theory techniques.

Contribution

It establishes the acyclicity of Thompson's group V and generalizes the homology computation to Higman-Thompson groups V_{n,r}, linking them to infinite loop space homology.

Findings

Thompson's group V is acyclic.

Homology of V_{n,r} matches that of a specific infinite loop space.

Homological stability with respect to r is established.

Abstract

We prove that Thompson's group $V$ is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman-Thompson groups $V_{n,r}$ with the homology of the zeroth component of the infinite loop space of the mod $n-1$ Moore spectrum. As $V = V_{2,1}$, we can deduce that this group is acyclic. Our proof involves establishing homological stability with respect to $r$, as well as a computation of the algebraic K-theory of the category of finitely generated free Cantor algebras of any type $n$.