A relationship between twisted conjugacy classes and the geometric invariants $Ω^n$

A group $G$ is said to have the property $R_\infty$ if every automorphism $ϕ\in {\rm Aut}(G)$ has an infinite number of $ϕ$-twisted conjugacy classes. Recent work of Gonçalves and Kochloukova uses the $Σ^n$ (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 $Ω^n$ invariants, analogous to $Σ^n$, 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 $Ω^n$ invariants can be used to determine the $R_{\infty}$ property while the $Σ^n$ invariants techniques cannot.

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