The root class residuality of the tree product of groups with amalgamated retracts
According to \textit{K. W. Gruenberg} [Proc. Lond. Math. Soc. (3) 7, 29--62 (1957; Zbl 0077.02901)], a non trivial class \(\mathcal{K}\) of groups is called a root class whenever \(\mathcal{K}\) is closed under subgroups and direct products of finitely many factors, as well as it satisfies the following condition: If \(1 \leq Z \leq Y \leq X\) is subnormal series of \(X\) whose factors \(X/Y\) and \(Y/Z\) lie in \(\mathcal{K}\) then \(X\) has a normal subgroup \(T\) such that \(T \leq Z\) and \(X/T \in \mathcal{K}\). By an equivalent definition (see [ Comm. Algebra, vol. 43, 856-860 (2015). ]) it follows that the intersection of root classes yields a root class. It is known (see [\textit{D. N. Azarov} and \textit{E. A. Tumanova}, ``On the residuality of generalized free products by root classes of groups,'' Nauch. Tr. Ivanovsk. Gos. Univ. 6, 29--42 (2008)], with generalizations in [\textit{E. A. Tumanova}, ``On residuality of generalized free products by root classes of groups,'' Model. Anal. Inform. Sist. 20, No. 1, 133--137 (2013);\textit{E. V. Sokolov} and \textit{E. A. Tumanova}, Sib. Math. J. 57, No. 1, 135--144 (2016; Zbl 1342.20035); translation from Sib. Mat. Zh. 57, No. 1, 171--185 (2016)]) that, for every root class \(\mathcal{K}\), the free product of two residually \(\mathcal{K}\)-groups with amalgamated subgroups which are a retract in the factor groups is again a residually \(\mathcal{K}\)-group. Recall that a subgroup \(Y\) of a group \(X\) is called a retract whenever \(X\) has a normal subgroup \(Z\) with \(X=YZ\) and \(Y\cap Z=1\). Equivalently \(Y\) is a retract in \(X\) if and only if \(X\) is the split extension of some group \(Z\) by \(Y\). In the present paper, a serious generalization of the result above is obtained. Theorem A. (Theorem 2 in the paper) Consider an arbitrary root class \(\mathcal{K}\) of groups, a tree \(T=(V,E)\), and the tree product \(G\) of groups \(A_{v} \) for \(v\in V\) with subgroups \(H_{vw}\) and \(H_{wv}\) amalgamated by isomorphisms \(\varphi _{ev}\) and \(\varphi _{ew}\) for \(e=\{v,w\}\in E\). If all \(A_{v}\) are residually \(\mathcal{K}\)-groups and, for each edge \(\{v,w\}\in E\), the subgroup \(H_{vw}\) is a retract in \(A_{v}\) and the subgroup \(H_{wv}\) is a retract in \(A_{w}\) then \(G\) is a residually \(\mathcal{K}\)-group. Let \((V,E)\) be an undirected graph with vertex set \(V\) and edge set \(E\), then it is well-known how Artin ( and the corresponding Coxeter) groups can be defined using the structure of the graph \((V,E)\). If the graph \((V,E)\) is a tree, then we can speek of the tree structure of Artin (Coxeter) groups. Here, characterizations for two-generator Artin (Coxeter) groups are given to be residually \(\mathcal{K}\)-groups, for any root class \(\mathcal{K}\) (Propositions 11 and 12 in the paper). After, using subdivision of edges of a tree it is showed that each Artin (Coxeter) group, with tree structure, can be considered as a tree product of groups where the edge (amalgamated) subgroups are retracts in the corresponding vertex groups. Therefore, applying Theorem A, a characterization for the \(\mathcal{K}\) residuality for Artin (Coxeter) groups with tree structure is obtained (Corollaries 1 and 2 in the paper). Another application of Theorem A is to obtain a sufficient condition for the \(\mathcal{K}\)-residuality of the \(HNN\)-extension of a residually \(\mathcal{K}\)-group. Theorem B. (Theorem 3 in the paper) Consider a root class \(\mathcal{K}\) of groups containing at least one nonperiodic group and the \(HNN\)-extension \(X\,=\,\langle \,B,\,t \mid t^{-1}Ht\,=\,K, \varphi \,\rangle \) of a residually \(\mathcal{K}\)-group \(B\) with subgroups \(H,\,K\leq B\) related by some isomorphism \(\varphi :H\,\longrightarrow\,K\). If \(H\) and \(K\) are retracts in \(B\), then \(X\) is a residually \(\mathcal{K}\)-group. Special but interesting corollaries of this theorem are obtained.
scientific article; zbMATH DE number 7137342
scientific article