A relationship between twisted conjugacy classes and the geometric invariants ${\mathrm{\Omega }}^n$

A group $G$ is said to have the property $R_{\mathrm{\infty }}$ if every automorphism $\varphi \in \mathrm{A}\mathrm{u}\mathrm{t}(G)$ has an infinite number of $\varphi$-twisted conjugacy classes. Recent work of Gon\c{c}alves and Kochloukova uses the ${\mathrm{\Sigma }}^n$ (Bieri-Neumann-Strebel-Renz) invariants to show the $R_{\mathrm{\infty }}$ property for a certain class of groups, including the generalized Thompson’s groups $F_{n,0}$. In this paper, we make use of the ${\mathrm{\Omega }}^n$ invariants, analogous to ${\mathrm{\Sigma }}^n$, to show $R_{\mathrm{\infty }}$ for certain finitely generated groups. In particular, we give an alternate and simpler proof of the $R_{\mathrm{\infty }}$ property for BS(1,n). Moreover, we give examples for which the ${\mathrm{\Omega }}^n$ invariants can be used to determine the $R_{\mathrm{\infty }}$ property while the ${\mathrm{\Sigma }}^n$ invariants techniques cannot.