Finite subgroups of the profinite completion of good 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).