Actions of arithmetic groups on homology spheres and acyclic homology manifolds

Abstract

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary $p$-groups in the groups concerned. In some cases, including ${\mathrm{Sp}}(2n,\mathbb Z )$, the bounds we obtain are sharp: if $X$ is a generalized $\mathbb Z /3$-homology sphere of dimension less than $2n-1$ or a $\mathbb Z /3$-acyclic $\mathbb Z /3$-homology manifold of dimension less than $2n$, and if $n\ge 3$, then any action of ${\mathrm{Sp}}(2n,\mathbb Z )$ by homeomorphisms on $X$ is trivial; if $n=2$, then every action of ${\mathrm{Sp}}(2n,\mathbb Z )$ on $X$ factors through the abelianization of ${\mathrm{Sp}}(4,\mathbb Z )$, which is $\mathbb Z /2$.