One-relator quotients of graph products. (Q2852581)

Let \(\Gamma\) be a graph with vertex set \(V\) and edge set \(E\). The right-angled Artin group associated to \(\Gamma\) is defined by NEWLINE\[NEWLINEA_\Gamma:=\langle V\mid [u,v]=1,\;\forall (u,v)\in E\rangle.NEWLINE\]NEWLINE A graph \(\Gamma\) is called \textit{starred} if it is finite and has no incidence of full subgraphs isomorphic to either \(C_4\), the cycle of length 4, or \(L_3\), the line of length 3. Connected starred graphs have a node. A node is a vertex which is adjacent to every other vertex. A right-angled Artin group is called starred if its defining graph is starred.NEWLINENEWLINE The authors show: Theorem 1.2. Let \(A_\Gamma\) be a starred right-angled Artin group and \(g\in A_\Gamma\). Let \(N\) be the set of nodal vertices in \(\Gamma\). Let \(G:=A_\Gamma/\langle\langle g\rangle\rangle\), the quotient of \(A_\Gamma\) by the normal closure of \(g\). ThenNEWLINENEWLINE (1) the word problem is solvable in \(G\),NEWLINENEWLINE (2) if \(U\subset N\) and \(g\notin A_{\Gamma_U}\), the \(A_{\Gamma_U}\) naturally embeds in \(G\),NEWLINENEWLINE (3) if \(U\) spans a sub-star and \(g\) is not conjugate to an element of \(A_{\Gamma_U}\) then \(A_{\Gamma_U}\) naturally embeds in \(G\).NEWLINENEWLINE The authors also show that the word problem is solvable for one-relator quotients of graph products over starred graphs with poly-(infinite cyclic) vertex groups.NEWLINENEWLINE At last there are two Theorems showing generalizations of the Freiheitssatz in case of direct products and they solve the membership problem in that case.