A group G is said to have property R ∞ R if, for every automorphism ℙ ⋯ Aut (G), the cardinality of the set of φ-twisted conjugacy classes is infinite. Many classes of groups are known to have this pr..

A group G is said to have property R ∞ R if, for every automorphism ℙ ⋯ Aut (G), the cardinality of the set of φ-twisted conjugacy classes is infinite. Many classes of groups are known to have this property. However, very few examples are known for which R ∞ R is geometric, i.e., if G has property R ∞ R, then any group quasi-isometric to G also has property R ∞ R. In this paper, we give examples of groups and conditions under which R ∞ R is preserved under commensurability. The main tool is to employ the Bieri-Neumann-Strebel invariant.