Actions of arithmetic groups on homology spheres and acyclic homology manifolds

TL;DR

This paper investigates the limitations on how arithmetic groups with torsion can act on acyclic manifolds and homology spheres, establishing bounds based on elementary p-groups and identifying cases where these bounds are sharp.

Contribution

It provides new lower bounds on the dimensions for group actions and characterizes when these actions are trivial or factor through abelianizations, especially for Sp(2n,Z).

Findings

Bounds are sharp for Sp(2n,Z) actions on certain homology spheres.

Actions of Sp(2n,Z) are trivial below dimension 2n-1 for n≥3.

For n=2, actions factor through Z/2, the abelianization of Sp(4,Z).

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 Sp(2n,Z), the bounds we obtain are sharp: if X is a generalized Z/3-homology sphere of dimension less than 2n-1 or a Z/3-acyclic Z/3-homology manifold of dimension less than 2n, and if n \geq 3, then any action of Sp(2n,Z) by homeomorphisms on X is trivial; if n = 2, then every action of Sp(2n,Z) on X factors through the abelianization of Sp(4,Z), which is Z/2.