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

Abstract

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

13 pages