An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements

Kim, Djun Maximilian; Rolfsen, Dale

doi:10.4153/cjm-2003-034-2

Cited by 22 publications

(25 citation statements)

References 22 publications

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“…(ii) (Bardakov [2], Kim and Rolfsen [39]) Let g and h be pure braids in B(A r ) such that g k = h k for some nonzero integer k. Then g and h are equal. (iii) (Lee and Lee [41]) Let G be one of the braid groups of types A r , B r , A r−1 and C r−1 .…”

Section: Uniqueness Of Roots Up To Conjugacymentioning

confidence: 98%

Braid groups of imprimitive complex reflection groups

Corran

Lee

2015

Journal of Algebra

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We obtain new presentations for the imprimitive complex reflection groups of type $(de,e,r)$ and their braid groups $B(de,e,r)$ for $d,r \ge 2$. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group $B(de,e,r)$ is a semidirect product of the braid group of affine type $\widetilde A_{r-1}$ and an infinite cyclic group. Elements of $B(de,e,r)$ are visualized as geometric braids on $r+1$ strings whose first string is pure and whose winding number is a multiple of $e$. We classify periodic elements, and show that the roots are unique up to conjugacy and that the braid group $B(de,e,r)$ is strongly translation discrete.Comment: published versio

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Section: Uniqueness Of Roots Up To Conjugacymentioning

confidence: 98%

Braid groups of imprimitive complex reflection groups

Corran

Lee

2015

Journal of Algebra

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“…Let π : B n → Sym n be the natural epimorphism from the braid group B n into the symmetric group Sym n . Then the kernel of π, denoted by PB n and called the pure braid group on n strands, is biorderable (see [24]). …”

Section: Genus 0 Surfacesmentioning

confidence: 99%

Roots in the mapping class groups

Bonatti

Paris

2008

Proceedings of the London Mathematical Society

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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...

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“…Observe that pure braid groups have the unique root property: they are biorderable by Kim and Rolfsen [19] and biorderable groups have the unique root property. It is well known that each of the braid groups (a.k.a.…”

Section: Some Group-theoretic Problemsmentioning

confidence: 97%