Finite subgroups of the profinite completion of good groups

TL;DR

This paper investigates the structure of finite subgroups within the profinite completion of certain good groups, establishing a bijective correspondence between conjugacy classes of finite p-subgroups in the group and its completion, with applications to geometric group theory.

Contribution

It proves a bijective correspondence between finite p-subgroups of a good group and its profinite completion, extending to solvable subgroups under certain conditions, with applications to hyperelliptic mapping class groups.

Findings

Bijective correspondence between conjugacy classes of finite p-subgroups in G and G.

Centralizers and normalizers in G are closures of those in G.

Applications to hyperelliptic mapping class groups and hyperbolic orbifold groups.

Abstract

Let $G$ be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism $G\hookrightarrow\hat{G}$ induces a bijective correspondence between conjugacy classes of finite $p$-subgroups of $G$ and those of its profinite completion $\hat{G}$. Moreover, we prove that the centralizers and normalizers in $\hat{G}$ 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).