On Brown's Problem, Poincare' models for the classifying spaces for proper actions and Nielsen Realization

TL;DR

This paper addresses the existence of cocompact proper topological models for classifying spaces of virtually torsionfree groups, solving related problems in the Poincaré category and providing new insights for hyperbolic groups.

Contribution

It offers a solution to Brown's problem in the Poincaré category under specific conditions, especially for hyperbolic groups, and advances understanding of classifying spaces for proper actions.

Findings

Solved Brown's problem in the Poincaré category for certain groups.

Established existence of models with zero-dimensional singular sets.

Provided new results for hyperbolic groups regarding classifying spaces.

Abstract

There is the problem, 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. It is related to Nielsen's Realization and to the problem of Brown, whether there is a d-dimensional model for the classifying space for proper actions, if the underlying group has virtually cohomological dimension d. Assuming that the expected manifold model has a zero-dimensional singular set, we solve the problem in the Poincar\'e category and obtain new results about Brown's problem under certain conditions concerning the underlying group, for instance if it is hyperbolic. In a sequel paper together with James Davis we will deal with this on the level of topological manifolds.