Intersection of centralizers in a partially commutative metabelian group (Q6160502)

Let \(\Gamma=\langle X;E\rangle\) be a finite undirected graph without loops with vertex set \(X=\{x_1,\dots, x_n\}\) and edge set \(E\subseteq X\times X\). A partially commutative group \(F(\Gamma, \mathfrak{M})\) of the variety \(\mathfrak{M}\) has presentation \(F(\Gamma, \mathfrak{M})=\langle X\mid x_ix_j=x_jx_i \text{ if }\{x_i,x_j\}\in E, \mathfrak{M}\rangle\). Let \(\mathfrak{A}^2\) denote the variety of all metabelian groups and recall that a clique of a graph is any of its complete subgraphs. The main aim of the paper is to prove the following result: Let \(G=F(\Gamma, \mathfrak{A}^2)\) and \(X=\{x_1,\dots, x_n\}\) the vertex set of the graph \(\Gamma\). Then for distinct vertices \(x, y\) of the graph \(\Gamma\), \(C(x)\cap C(y)\cap G'\) is trivial if and only if whenever \(x_{i_1},\dots, x_{i_m},x_{i_1}\) is a cycle, then \(\{x_{i_1},\dots, x_{i_m}\}\) is a clique. (Here \(C(x)\) denotes the centralizer of \(x\) in \(G\) and \(G'\) is the commutator subgroup.)