The genus of HNN-extensions. (Q2841687)

Let \(G\) be a group and \(\widehat G\) its profinite completion. In [\textit{F. Grunewald} and \textit{P. Zalesskii}, J. Algebra 326, No. 1, 130-168 (2011; Zbl 1222.20019)] it is introduced the notion of genus for a group \(G\) with respect to some family \(\mathfrak F\) of groups as a set of isomorphism classes of groups of the family having the same profinite completion: \(g(\mathfrak F,G)=\text{IsoClasses}(\{H\in\mathfrak F\mid\widehat G\cong\widehat H\})\).NEWLINENEWLINE In the above mentioned paper were formulated some problems concerning the notion of genus and the genus of an amalgamated free product of finite groups with respect to the class of virtually free groups was studied.NEWLINENEWLINE In the present paper the authors study the genus of HNN-extensions with finite base group with respect to the class of virtually free groups.NEWLINENEWLINE Let \(G=\langle t,K\mid\text{rel}(K),\;f(a)=t^{-1}at,\;a\in A\rangle\) be the HNN-extension of the finite group \(K\), where \(f\colon A\to B\) is an isomorphism between the subgroups \(A\) and \(B\) of \(K\).NEWLINENEWLINE Sufficient conditions (Theorem 1.1 ) on the subgroups \(A\) and \(B\) and on the isomorphism \(f\) are given for \(|g(\mathfrak{DF},G)|=1\), where \(\mathfrak{DF}\) is the class of virtually free groups.NEWLINENEWLINE Let \(f_i\colon A\to B\), \(i=1,2\), be two isomorphisms and \(G_i=\text{HNN}(K,A,f_i)\) the corresponding HNN-extensions. A characterization upon the isomorphisms \(f_i\) is given for \(G_1\cong G_2\) (Theorem 4.5). This enables the authors to obtain bounds for \(|g(\mathfrak{DF},G)|\) (Theorem 5.1).NEWLINENEWLINE Finally the authors study the notion of \(p\)-genus of an HNN-extension \(G=\text{HNN}(K,A,f)\) with base group \(K\) a finite \(p\)-group.NEWLINENEWLINE Let \(g_p(\mathfrak{DF},G)=\text{IsoClasses}(\{H\in\mathfrak{DF}\mid\widehat G_p\cong\widehat H_p\})\), where \(\widehat G_p\) is the pro-\(p\) completion of \(G\) provided that \(G\) is residually-\(p\).NEWLINENEWLINE An algorithmically checkable characterization (Theorem 7.1) is given when two HNN-extensions \(G_1=\text{HNN}(K,A,f_1)\) and \(G_2=\text{HNN}(K,A,f_2)\), with base group \(K\) a finite \(p\)-group, have isomorphic pro-\(p\) completions.NEWLINENEWLINE Also a calculation (Theorem 7.2) of \(|g_p(\mathfrak{DF},G)|\) is obtained.