Free actions of p-groups on products of lens spaces

Let p be an odd prime number. We prove that if (Z=p)r acts freely on a product of k equidimensional lens spaces, then r k. This settles a special case of a conjecture due to C. Allday. We also nd further restrictions on non-abelian p-groups acting freely on a product of lens spaces. For actions inducing a trivial action on homology, we reach the following characterization: A p-group can act freely on a product of k lens spaces with a trivial action on homology if and only if rk(G) k and G has the -extension property. The main technique is to study group extensions associated to free actions.