Ascending chains of finitely generated subgroups (Q342842)

This is a very nice paper showing that its author has good mathematical taste. The results of the paper split into two topics.NEWLINENEWLINE1. Let \(d(G)\) denote the minimal number of generators of a group \(G\) (if \(G\) is pro-\(p\) or profinite this means topological generators). For a (closed) subgroup \(H\) of a pro-\(p\) group \(G\), the root \(\sqrt{H}\) of \(H\) is the maximal subgroup of \(G\) containing \(H\) as an open subgroup. The first main result of the paper is the followingNEWLINENEWLINETheorem 1.2. Let \(\Gamma\) be a pro-\(p\) group such that for any finitely generated subgroup \(K\) of \(\Gamma\) NEWLINE\[NEWLINE\mathrm{inf}_{U\leq_o K} (d(U)-1)/[K:U]NEWLINE\]NEWLINE is positive. Then, for any non-trivial finitely generated subgroup \(H\) of \(\Gamma\) the commensurator \(\mathrm{Comm}_\Gamma(H)=\sqrt{H}\) and the action of any subgroup \(\sqrt{H}< L\leq\Gamma\) on \(L/H\) by the left multiplication is faithful.NEWLINENEWLINEThe methods of the proof work not only for pro-\(p\) groups and in fact give a uniform treatment of numerous results of this nature in the literature. In particular, the author deduces the finiteness of \([N_{\Gamma}(H):H]\) as a corollary, the fact proved for many groups of combinatorial and geometric nature (like free, surface, limit groups in both abstract and profinite, especially pro-\(p\) categories) by different methods. Note that \([N_{\Gamma}(H):H]< \infty\) implies in particular that \(\Gamma\) does not have (non-trivial) finitely generated normal subgroups.NEWLINENEWLINE2. The other theme is the study of a family of \(n\)-generated subgroups \(\mathcal F\) of a pro-\(p\) or abstract group \(G\). The author proves that \(\mathcal F\) has a maximal element by inclusion in two situations:NEWLINENEWLINE\(\bullet\) \(G\) is a pro-\(p\) group;NEWLINENEWLINE\(\bullet\) \(G\) is a group for which every subgroup with finitely generated profinite completion is itself finitely generated.