Twisted conjugacy and commensurability invariance

Abstract

A group $G$ is said to have property $R_{\infty }$ if for every automorphism $\phi \in \mathrm{Aut}(G)$, the cardinality of the set of $\phi$-twisted conjugacy classes is infinite. Many classes of groups are known to have such property. However, very few examples are known for which $R_{\infty }$ is {\it geometric}, i.e., if $G$ has property $R_{\infty }$ then any group quasi-isometric to $G$ also has property $R_{\infty }$. In this paper, we give examples of groups and conditions under which $R_{\infty }$ is preserved under commensurability. The main tool is to employ the Bieri-Neumann-Strebel invariant.

17 pages