Twisted conjugacy in big mapping class groups

Abstract

Let G be a group and φ be an automorphism of G. Two elements x,y of G are said to be φ-twisted conjugate if y=gxφ(g)−1 for some g∈G. A group G has the R∞-property if the number of φ-twisted conjugacy classes is infinite for every automorphism φ of G. In this paper, we prove that the big mapping class group MCG(S) possesses the R∞-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∞-property if and only if it satisfies the S∞-property.