Actions of arithmetic groups on homology
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.