Virtually cyclic dimension for 3-manifold groups

Abstract

Let $\Gamma$ be the fundamental group of a connected, closed, orientable $3$-manifold. We explicitly compute its virtually cyclic geometric dimension $\underline{\underline{\mathrm{gd}}}(\Gamma )$. Among the tools we use are the prime and JSJ decompositions of $M$, acylindrical splittings of groups, several push-out type constructions, as well as some Bredon cohomology computations.