Twisted conjugacy in generalized Thompson groups of type F

TL;DR

This paper proves that certain generalized Thompson groups have the $R_inite$-property, meaning they have infinitely many twisted conjugacy classes for every automorphism, which has implications in geometric group theory.

Contribution

The paper establishes that generalized Richard Thompson groups $F_n$ and $F(l,A,P)$ possess the $R_inite$-property, extending previous results to broader classes of groups.

Findings

Generalized Thompson groups have the $R_inite$-property.

Infinite twisted conjugacy classes exist for all automorphisms.

Results contribute to understanding automorphism dynamics in these groups.

Abstract

If $\phi$ is an automorphism of a group $G$ and $x,y\in G$, we say that $x$ and $y$ are $\phi$-twisted conjugates if there exists an $z\in G$ such that $y=z.x.\phi(z^{-1})$. This is an equivalence relation. If there are infinitely many $\phi$-twisted conjugacy classes for every automorphism $\phi$ of $G$ we say that $G$ has the $R_\infty$-property. We prove that the generalized Richard Thompson groups $F_n$ and $F(l,A,P)$ have the $R_\infty$-property.