Roots in the mapping class groups

Abstract

The purpose of this paper is the study of the roots in the mapping class groups. Let Σ be a compact oriented surface, possibly with boundary, let P be a finite set of punctures in the interior of Σ, and let M(Σ, P) denote the mapping class group of (Σ, P). We prove that, if Σ is of genus 0, then each f ∈ M(Σ) has at most one m-root for all m ≥ 1. We prove that, if Σ is of genus 1 and has non-empty 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 ≥ 2, then there exist f, g ∈ M(Σ, P) 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, P) has at most one m-root for all m ≥ 1. We prove that, however, if ∂Σ = ∅ and the genus of Σ is ≥ 2, then there exist two pseudo-Anosov elements f, g ∈ M(Σ) (explicitely 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. Furthermore, if the genus of Σ is ≡ 0 (mod 4), then we can take m = 2. Finally, we show that, if Γ is a pure subgroup of M(Σ, P) 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, P).AMS Subject Classification: Primary 57M99. Secondary 57N05, 57R30. IntroductionThroughout the paper Σ will denote a compact oriented surface, possibly with boundary, and P = {P 1 , . . . , P n } a finite collection of points, called the punctures, in the interior of Σ. We denote by Diff(Σ, P) the group of orientation preserving diffeomorphisms F : Σ → Σ which are the identity on a neighborhood of the boundary of Σ and such that F (P) = P. The mapping class group of (Σ, P) is defined to be the group M(Σ, P) = π 0 (Diff(Σ, P)) of isotopy classes of elements of Diff(Σ, P). Note that, in the above definition, one may replace Diff(Σ, P) by Homeo(Σ, P), the group of orientation preserving homeomorphisms F : Σ → Σ which are the identity on ∂Σ and such that F (P) = P (namely, M(Σ, P) = π 0 (Diff(Σ, P)) = π 0 (Homeo(Σ, P)). Note also that the hypothesis that the elements of Diff(Σ, P) restrict to the identity on a neighborhood of ∂Σ (and not only on ∂Σ) is especially needed when considering subsurfaces. Indeed, if Σ ′ is a subsurface of Σ such that P ∩ ∂Σ ′ = ∅, then the embedding Σ ′ ⊂ Σ determines an embedding Diff(Σ ′ , P ∩ Σ ′ ) → Diff(Σ, P) by extending each diffeomorphism F ∈ Diff(Σ ′ , P ∩ Σ ′ ) with the identity map outside Σ ′ , and this monomorphism determines a homomorphism M(Σ ′ , P ∩ Σ ′ ) → M(Σ, P) which is injective in most of the cases but not always (see [29]).If Σ = D is the standard disk and |P| = n, then M(Σ, P) is the braid group B n introduced by Artin [2], [3]. 1 Let Γ be a group, g ∈ Γ, and m ≥ 1. Define a m-root of g to be an element f ∈ Γ such that f m = g. If Γ = Z q , then each element g ∈ Γ has at most one m-root. The same result is true if Γ is free or, more genera...

Milnor open books of links of some rational surface singularities

“…We now use the fact that elements in mapping class groups of surfaces of genus one with nonempty boundary can have at most one m-root up to conjugation (cf. [6]) to deduce that ψ n may be written ψ n = (α 1 α 2 β) 2 δ 1 δ 2 · · · δ n−2 which indeed coincides with φ n above. Here we used the two-holed torus relation δ n−1 x = (α 1 α 2 β) 4 .…”

Periodic elements in Garside groups

“…If G is the braid group of a well-generated complex reflection group and g and h are periodic elements, then g and h are conjugate by Bessis [4]. For a study of roots in mapping class groups, see [13]. Theorem 3.9 shows that if G is a Garside group and h is a power of a Garside element ∆, then (1) implies that g and h are conjugate.…”

Approximate roots of pseudo-Anosov diffeomorphisms

“…Denote by f u m , f s m the unstable and stable laminations of Ψ m , which we assume are parametrized by the L 1 norm 2m 2 …”

“…It may also be useful in the search for genuine roots of pseudo Anosov diffeomorphisms, which have been the subject of several recent studies e.g. [2], [6]. In this article, we shall prove an analogue of the Root Conjecture in which the Teichmüller topology on T is replaced by the heights-widths topology.…”