spheres and acyclic homology Manifolds.

**M.R. Bridson, F. Grunewald and K. Vogtmann**

* Submitted ito Math Zeitschrift August 2012. In final form 12 June 2013.*

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, 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.

10 pages, no figures.