Twisted conjugacy in big mapping class groups

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 is an element of G. A group G has the R infinity-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 infinity-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 infinity-property if and only if it satisfies the S infinity-property. (c) 2022 Elsevier B.V. All rights reserved.