$R_{\infty }$-property for groups commensurable to nilpotent quotients of RAAGs

Abstract

Let $G$ be a group and $\phi$ an automorphism of $G$. Two elements $x,y\in G$ are said to be $\phi$-conjugate if there exists a third element $z\in G$ such that $zx\phi (z{)}^{-1}=y$. Being $\phi$-conjugate defines an equivalence relation on $G$. The group $G$ is said to have the $R_{\infty }$-property if all its automorphisms $\phi$ have infinitely many $\phi$-conjugacy classes. For finitely generated torsion-free nilpotent groups, the so-called Mal'cev completion of the group is a useful tool in studying this property. Two groups have isomorphic Mal'cev completions if and only if they are abstractly commensurable. This raises the question whether the $R_{\infty }$-property is invariant under abstract commensurability within the class of finitely generated torsion-free nilpotent groups. We show that the answer to this question is negative and provide counterexamples within a class of 2-step nilpotent groups associated to edge-weighted graphs. These groups are commensurable to 2-step nilpotent quotients of right-angled Artin groups.

25 pages