On Nielsen realization and manifold models for classifying spaces

We consider the problem of whether, for a given virtually torsionfree discrete group $\mathrm{\Gamma }$, there exists a cocompact proper topological $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }\mathrm{/}\pi$ is a finite cyclic group of odd order.