Linear Groups, Conjugacy Growth, and Classifying Spaces for Families of Subgroups

2017/04/18 by Timm von Puttkamer, von Puttkamer, Timm, Xiaolei Wu +1
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1704.05304

openalex publication_date 2017/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a group G and a family of subgroups F, we consider its classifying space E_{\mathcal F}G with respect to F. When \mathcal F = VCyc 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_VCyc(G) whose 0-skeleton is finite.

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