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This research addresses whether a cocompact proper topological manifold exists for virtually torsion-free discrete groups under specific conditions.

March 1, 2026

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

Key Points

This research addresses whether a cocompact proper topological manifold exists for virtually torsion-free discrete groups under specific conditions.
Exploration of the relationship between cocompact proper topological Γ-manifolds and classifying spaces.
Examination of conditions under which the manifold model has a zero-dimensional singular set.
Analysis within the Poincaré category.
Demonstrated solutions to Brown's problem for specific classes of groups.
Identified conditions under which hyperbolic groups yield successful outcomes.
Provided insights into Nielsen's Realization problem in this context.

Abstract

There is the problem, whether for a given virtually torsionfree discrete group Γ there exists a cocompact proper topological Γ-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 Poincaré 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.

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

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