Finite subgroups of the profinite completion of good groups (Q6663908)

Let \(G\) be a residually finite group,\(\widehat{G}\) its profinite completion and \(\iota : G\rightarrow \widehat{G}\) the natural monomorphism. The map \(\iota\) induces a map \(\iota_{c}\) from the set of conjugacy classes of finite subgroups of \(G\) to the set of conjugacy classes of finite subgroups of \(\widehat{G}\).\N\NIn the paper under review, the authors prove, under the assumption that \(G\) is good (see Definition 1.1), that \(\iota_{c}\) induces a bijective correspondence between conjugacy classes of finite \(p\)-subgroups of \(G\) and those of \(\widehat{G}\). Moreover, they prove that the centralizers and normalizers in \(\widehat{G}\) of finite \(p\)-subgroups of \(G\) are the closures of the respective centralizers and normalizers in \(G\). They also prove that, with more restrictive hypotheses, the same result holds for finite solvable subgroups of \(G\).\N\NIn the last section, the authors give some applications of the results achieved to hyperelliptic mapping class groups and virtually compact special toral relatively hyperbolic groups.