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

Maarten Lathouwers and Thomas Witdouck

Mathematische Nachrichten, 2025, vol. 298, issue 2, 602-616

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

Date: 2025
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https://doi.org/10.1002/mana.202400154

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