On Nielsen realization and manifold models for classifying spaces

Abstract

We consider the problem of whether, for a given virtually torsionfree discrete group $\Gamma$, there exists a cocompact proper topological $\Gamma$-manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. This problem is related to Nielsen Realization. We will make the assumption that the expected manifold model has a zero-dimensional singular set. Then we solve the problem in the case, for instance, that $\Gamma$ contains a normal torsionfree subgroup $\pi$ such that $\pi$ is hyperbolic and $\pi$ is the fundamental group of an aspherical closed manifold of dimension greater or equal to five and $\Gamma /\pi$ is a finite cyclic group of odd order.

41 pages. The final, final version, to appear in the Transactions of the American Mathematical Society