The homology of the Higman–Thompson groups

Abstract

We prove that Thompson’s group \mathrm {V} is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman–Thompson groups \mathrm {V}_{n,r} with the homology of the zeroth component of the infinite loop space of the mod n-1 Moore spectrum. As \mathrm {V}=\mathrm {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.