Separability properties of automorphisms of graph products of groups. (Q2796975)

Let \(\Gamma\) be a simplicial graph with set of vertices \(V\Gamma\) and set of edges \(E\Gamma\). Suppose that \(\mathcal G=\{G_v\mid v\in V\Gamma\}\) is a family of groups, then the graph product \(\Gamma\mathcal G\) of the family \(\mathcal G\) with respect to the graph \(\Gamma\) is defined to be the quotient of the free product \(*_{v\in V\Gamma }G_v\) modulo the relations of the form \(g_ug_v=g_vg_u\) for every \(g_u\in G_u\), \(g_v\in G_v\) whenever \(\{u,v\}\in E\Gamma\). Extreme cases of graph products are the free product \(*_{v\in V\Gamma}G_v\) and the direct product \(\bigtimes_{v\in V\Gamma}G_v\). In the case where all the vertex groups are infinite cyclic we have the right angled Artin groups and in the case where all the vertex groups are cyclic of order two we have the right angled Coxeter groups.NEWLINENEWLINE An interesting problem is to decide when the group \(\mathrm{Out}(G)\) of the outer automorphisms of a graph product \(G=\Gamma\mathcal G\) is residually finite (\(\mathcal{RF}\)). In [\textit{A. Minasyan}, Groups Geom. Dyn. 6, No. 2, 335-388 (2012; Zbl 1280.20040)] and in [\textit{R. Charney} and \textit{K. Vogtmann}, Bull. Lond. Math. Soc. 41, No. 1, 94-102 (2009; Zbl 1244.20036)], it was proved that the outer automorphism group of a finitely generated right angled Artin group is \(\mathcal{RF}\) and in [\textit{P.-E. Caprace} and \textit{A. Minasyan}, Ill. J. Math. 57, No. 2, 499-523 (2013; Zbl 1306.20043)] that the outer automorphism group of a finitely generated right angled Coxeter group is \(\mathcal{RF}\).NEWLINENEWLINE In the paper under consideration the author studies the residual finiteness of the outer automorphisms of graph products.NEWLINENEWLINE Before stating the main results some terminology and some known results.NEWLINENEWLINE Let \(G\) be a group. The map \(\varphi\colon G\to G\) is pointwise inner if \(\varphi(g)\) is conjugate to \(g\) for every \(g\in G\). A group \(G\) has Grossman's property if every pointwise inner automorphism of \(G\) is inner. In [\textit{E. K. Grossman}, J. Lond. Math. Soc., II. Ser. 9, 160-164 (1974; Zbl 0292.20032)], it was proved that if the finitely generated group \(G\) is conjugacy separable (\(\mathcal{CS}\)) and satisfies Grossman's property, then its outer automorphism group \(\mathrm{Out}(G)\) is \(\mathcal{RF}\). These conditions are sufficient, but not necessary, for example there is a torsion-free polycyclic group with non-inner pointwise inner automorphisms with \(\mathcal{RF}\) outer automorphism group.NEWLINENEWLINE Let \(\mathcal C\) be a family of groups. A group \(G\) is said to be \(\mathcal C\)-inner automorphism separable (\(\mathcal C\)-IAS) if for every \(\varphi\in\Aut(G)\setminus\mathrm{Inn}(G)\) there exists a characteristic subgroup \(K\) of \(G\) such that the quotient \(G/K\) belongs to \(\mathcal C\) and the induced \(\overline\varphi\) automorphism in \(\Aut(G/K)\) does not belong to \(\mathrm{Inn}(G/K)\). In case where \(\mathcal C\) is the class of all finite groups then the group \(G\) simply is said to be IAS and it is obvious that, in this case, the group \(\mathrm{Out}(G)\) is \(\mathcal{RF}\).NEWLINENEWLINE Let \(\Gamma \) be a graph. A subset \(U\subseteq V\Gamma\) is coneless if the vertices in \(U\) do not have a common neighbor. A vertex \(v\in V\Gamma\) is central in \(\Gamma\) if it is adjacent to all the vertices of \(\Gamma\).NEWLINENEWLINE Proposition 1. (Proposition 2.2 in the paper) Let \(\mathcal C\) be a class of finite groups which is closed for subgroups and forming direct products. Let \(A,B\) be finitely generated \(\mathcal C\)-IAS residually-\(\mathcal C\) groups. Then the direct product \(A\times B\) is \(\mathcal C\)-IAS.NEWLINENEWLINE Theorem 1. (Theorem 1.1 in the paper) Let \(\Gamma\) be a graph and suppose that there is \(U\subseteq V\Gamma\) such that \(|U|<\infty\) and \(U\) is coneless. Let \(\mathcal G=\{G_v\mid v\in V\Gamma\}\) be a family of nontrivial groups and let \(G=\Gamma\mathcal G\) be the graph product of \(\mathcal G\) with respect to \(\Gamma\). Then every pointwise inner endomorphism of \(G\) is an inner automorphism. In particular, the group \(\Gamma\mathcal G\) has Grossman's property.NEWLINENEWLINE Corollary 1. (Corollary 1.2 in the paper) Let \(\Gamma\) be a finite graph and let \(\mathcal G=\{G_v\mid v\in V\Gamma\}\) be a family of nontrivial groups. The group \(G=\Gamma\mathcal G\) has Grossman's property if and only if all vertex groups correponding to central vertices of \(\Gamma\) have Grossman's property.NEWLINENEWLINE Proposition 2. (Proposition 6.2 in the paper) Let \(\Gamma\) be a finite simplicial graph without central vertices and let \(\mathcal G=\{G_v\mid v\in V\Gamma\}\) be a family of nontrivial finitely generated residually-\(\mathcal C\) groups. Then the group \(\Gamma\mathcal G\) is \(\mathcal C\)-IAS.NEWLINENEWLINE In case where the class \(\mathcal C\) is all the finite groups, it is proved that the group \(\Gamma\mathcal G\) is IAS, consequently the group \(\mathrm{Out}(\Gamma\mathcal G)\) is \(\mathcal{RF}\) (Theorem 1.3 in the paper).NEWLINENEWLINE This result covers the results in [\textit{A. Minasyan} and \textit{D. Osin}, Trans. Am. Math. Soc. 362, No. 11, 6079-6103 (2010; Zbl 1227.20041)] where it is proved that the group of the outer automorphisms of a free product of finitely generated \(\mathcal{RF}\) groups is \(\mathcal{RF}\).NEWLINENEWLINE It is proved that virtually polycyclic groups are IAS (Lemma 3.5 in the paper). Therefore, if the family \(\mathcal G=\{G_v\mid v\in V\Gamma\}\) consists of virtually polycyclic groups, then the group \(\mathrm{Out}(\Gamma\mathcal G)\) is \(\mathcal{RF}\) (Corollary 1.5 in the paper).NEWLINENEWLINE Relevant results are obtained if the class \(\mathcal C\) is all the finite \(p\)-groups, \(p\) prime. (Theorem 1.6 and Corollary 1.7 in the paper).NEWLINENEWLINE For the Torelli group \(\mathrm{Tor}(\Gamma\mathcal G)\), in case where the family \(\mathcal G=\{G_v\mid v\in V\Gamma\}\) consists of residually torsion-free nilpotent groups, it is proved that it is residually \(p\)-finite for every prime \(p\) and it is bi-orderable (Theorem 1.8 in the paper).NEWLINENEWLINE The paper concludes with two interesting questions.NEWLINENEWLINE Question 1. Let \(A,B\) be finitely generated \(\mathcal{RF}\) groups such that \(\mathrm{Out}(A),\mathrm{Out}(B)\) are \(\mathcal{RF}\). Is \(\mathrm{Out}(A\times B)\) \(\mathcal{RF}\)?NEWLINENEWLINE Question 2. Is there a finitely generated \(\mathcal{RF}\) group \(G\) such that \(\mathrm{Out}(G)\) is an infinite \(\mathcal{RF}\) group but \(G\) is not IAS?