Linear groups, conjugacy growth, and classifying spaces for families of 
subgroups

Puttkamer, Timm von; Wu, Xiaolei

doi:10.1093/imrn/rnx215

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Linear groups, conjugacy growth, and classifying spaces for families of subgroups

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Puttkamer, Timm von
Max Planck Institute for Mathematics, Max Planck Society;

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Wu, Xiaolei
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1093/imrn/rnx215
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arXiv:1704.05304.pdf
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Citation

Puttkamer, T. v., & Wu, X. (2019). Linear groups, conjugacy growth, and classifying spaces for families of subgroups. International Mathematics Research Notices, 2019(10), 3130-3168. doi:10.1093/imrn/rnx215.

Cite as: https://hdl.handle.net/21.11116/0000-0004-7266-9

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.