Finite subgroups of the profinite completion of good groups

2024/06/12 by Marco Boggi, Boggi, Marco, Pavel Zalesskii +1 · 1 citation
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2406.08639

openalex publication_date 2024/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism G\hookrightarrowG induces a bijective correspondence between conjugacy classes of finite p-subgroups of G and those of its profinite completion G. Moreover, we prove that the centralizers and normalizers in G of finite p-subgroups of G are the closures of the respective centralizers and normalizers in G. With somewhat more restrictive hypotheses, we prove the same results for finite solvable subgroups of G. In the last section, we give a few applications of this theorem to hyperelliptic mapping class groups and virtually compact special toral relatively hyperbolic groups (these include fundamental groups of 3-orbifolds and of uniform standard arithmetic hyperbolic orbifolds).

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