The homology of the Higman–Thompson groups

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.