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Mathematics, Group Theory, Algebraic Topology
Abstract:
Given a group $G$ and a family of subgroups $\mathcal{F}$, we consider its
classifying space $E_{\mathcal F}G$ with respect to $\mathcal{F}$. When
$\mathcal F = \mathcal{VC}yc$ is the family of virtually cyclic subgroups, Juan-Pineda and Leary conjectured that a group admits a finite model for this classifying space if and only if it is virtually cyclic. By establishing a connection to conjugacy growth we can show that this conjecture holds for linear groups. We investigate a similar question that was asked by Lück-Reich-Rognes-Varisco for the family of cyclic subgroups. Finally, we construct finitely generated groups that exhibit wild inner automorphims but which admit a model for $E_{\mathcal{VC}yc}(G)$ whose 0-skeleton is finite.
