Twisted Conjugacy in Big Mapping Class Groups

Abstract

Let $G$ be a group and $\phi$ be an automorphism of $G$. Two elements $x,y$ of $G$ are said to be $\phi$-twisted conjugate if $y=gx\phi (g{)}^{-1}$ for some $g\in G$. A group $G$ has the $R_{\infty }$-property if the number of $\phi$-twisted conjugacy classes is infinite for every automorphism $\phi$ 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.

arXiv comment17 pages. To appear in Topology and its Applications