The R∞$R_\infty$‐property and commensurability for nilpotent groups

Abstract

For finitely generated torsion‐free nilpotent groups, the associated Mal'cev Lie algebra of the group is used frequently when studying the ‐property. Two such groups have isomorphic Mal'cev Lie algebras if and only if they are abstractly commensurable. We show that the ‐property is not invariant under abstract commensurability within the class of finitely generated torsion‐free nilpotent groups by providing counterexamples within a class of 2‐step nilpotent groups associated to edge‐weighted graphs. These groups are abstractly commensurable to 2‐step nilpotent quotients of right‐angled Artin groups.