  A group $G$ is said to have property $R_{\infty}$ if for every automorphism
$\varphi \in {\rm Aut}(G)$, the cardinality of the set of $\varphi$-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.
