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

TL;DR

This paper explores the connection between twisted conjugacy classes and geometric invariants, using $\Omega^n$ invariants to establish the $R_ty$ property for certain groups, offering simpler proofs and new examples.

Contribution

It introduces the use of $\Omega^n$ invariants to analyze the $R_ty$ property, providing an alternative approach and new examples where these invariants are effective.

Findings

Proves $R_ty$ for BS(1,n) using $\Omega^n$ invariants.

Identifies cases where $\Omega^n$ invariants succeed where $\Sigma^n$ do not.

Provides simpler proofs for known $R_ty$ properties.

Abstract

A group $G$ is said to have the property $R_\infty$ if every automorphism $\phi \in {\rm Aut}(G)$ has an infinite number of $\phi$-twisted conjugacy classes. Recent work of Gon\c{c}alves 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.