Twisted conjugacy and commensurability invariance

Abstract

Abstract A group 𝐺 is said to have property R ∞ R_{\infty} if, for every automorphism φ ∈ Aut ⁢ ( G ) \varphi\in\mathrm{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_{\infty} is geometric, i.e., if 𝐺 has property R ∞ R_{\infty} , then any group quasi-isometric to 𝐺 also has property R ∞ R_{\infty} . In this paper, we give examples of groups and conditions under which R ∞ R_{\infty} is preserved under commensurability. The main tool is to employ the Bieri–Neumann–Strebel invariant.