Twisted Conjugacy in Big Mapping Class Groups

TL;DR

This paper investigates the twisted conjugacy classes in big mapping class groups, establishing conditions under which these groups exhibit the $R_{ extinfty}$-property, and linking it to the $S_{ extinfty}$-property.

Contribution

It proves that big mapping class groups have the $R_{ extinfty}$-property under certain conditions and connects this property to the $S_{ extinfty}$-property.

Findings

Big mapping class groups have the $R_{ extinfty}$-property under specific conditions.

The $R_{ extinfty}$-property is equivalent to the $S_{ extinfty}$-property in these groups.

The results extend understanding of conjugacy properties in infinite-type surface groups.

Abstract

Let $G$ be a group and $\varphi$ be an automorphism of $G$. Two elements $x, y$ of $G$ are said to be $\varphi$-twisted conjugate if $y=gx\varphi(g)^{-1}$ for some $g\in G$. A group $G$ has the $R_{\infty}$-property if the number of $\varphi$-twisted conjugacy classes is infinite for every automorphism $\varphi$ of $G$. In this paper we prove that the big mapping class group $MCG(S)$ possesses the $R_{\infty}$-property under some suitable conditions on the infinite-type surface $S$. As an application we also prove that the big mapping class group possesses the $R_\infty$-property if and only if it satisfies the $S_{\infty}$-property.