Finite subgroups of the profinite completion of good groups

Abstract

Let be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism induces a bijective correspondence between conjugacy classes of finite ‐subgroups of and those of its profinite completion . Moreover, we prove that the centralizers and normalizers in of finite ‐subgroups of are the closures of the respective centralizers and normalizers in . With somewhat more restrictive hypotheses, we prove the same results for finite solvable subgroups of . 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).