Dimension of Virtually Cyclic Classifying Spaces for Certain Geometric Groups

Given a connected, oriented, closed 3-manifold M, we construct models for E_VCΓ, the classifying space of Γ = π₁(M) with isotropy in the virtually cyclic subgroups; we also compute the smallest possible geometric dimension for E_VCΓ, pointing out in which cases the models are larger than necessary.

This is done by decomposing M using the prime and JSJ decompositions; the resulting manifolds are either closed and geometric or compact with geometric interior by Thurston's Geometrization Conjecture. We develop a pushout construction of models for virtually cyclic classifying spaces of fundamental groups of Seifert fiber spaces with base orbifold modeled on H², then (using a pushout method of Lafont and Ortiz) we combine these with known models for the remaining pieces to obtain a model for E_VCΓ. These models are then analyzed using Bredon cohomology theory to see if they are of the smallest possible dimension.