Roots in the mapping class groups

The purpose of this paper is to study the roots in the mapping class groups. Let Σ be a compact oriented surface, possibly with boundary, let be a finite set of punctures in the interior of Σ, and let ℳ (Σ, ) denote the mapping class group (relative to the boundary) of (Σ, ). We prove that if Σ is of genus 1 and has nonempty boundary, then each f ∈ ℳ (Σ) has at most one m ‐root up to conjugation for all m ⩾ 1. We prove that, however, if Σ is of genus at least 2, then there exist f , g ∈ ℳ (Σ, ) such that f 2 = g 2 , f is not conjugate to g , and none of the conjugates of f commutes with g . Afterwards, we focus our study on the roots of the pseudo‐Anosov elements. We prove that if ∂ Σ ≠ ∅, then each pseudo‐Anosov element f ∈ ℳ(Σ, ) has at most one m ‐root for all m ⩾ 1, but if ∂ Σ = ∅ then there exist two pseudo‐Anosov elements f , g ∈ ℳ (Σ) (explicitly constructed) such that f m = g m for some m ⩾ 2, f is not conjugate to g , and none of the conjugates of f commutes with g . Finally, we show that if Γ is a pure subgroup of ℳ (Σ, ) and f ∈ Γ, then f has at most one m ‐root in Γ for all m ⩾ 1. Note that there are finite‐index pure subgroups in ℳ(Σ, ).