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

For finitely generated torsion-free nilpotent groups, the associated Mal'cev Lie algebra of the group is used frequently when studying the $R_{\infty }$-property. Two such groups have isomorphic Mal'cev Lie algebras if and only if they are abstractly commensurable. We show that the $R_{\infty }$-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.