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.
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