  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 $n$.
