A Partial Order on the Symmetric Group and New K(π, 1)'s for the Braid Groups

Brady, Thomas A.

doi:10.1006/aima.2001.1986

Cited by 75 publications

(127 citation statements)

References 9 publications

(26 reference statements)

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“…The resulting quotient is a complex we call the Brady-Krammer complex BK n , since it was discovered independently by Tom Brady and Daan Krammer. What Brady and Krammer proved in [12] and [31], respectively, was that this procedure results in a complex whose fundamental group is the braid group B n and whose universal cover is contractible. In other words, they proved the following result:…”

Section: Braid Groupsmentioning

confidence: 98%

“…Instead of giving a complete proof we write down the isomorphism and omit the details. The interested reader can find a complete proof in [12]. Given a noncrossing partition σ, we convert it into a permutation π(σ) by writing each block as a disjoint cycle.…”

Section: Symmetric Groupsmentioning

confidence: 99%

“…First defined and studied by Germain Kreweras in 1972 [33], it caught the imagination of combinatorialists beginning in the 1980s [20], [21], [22], [23], [29], [37], [39], [40], [45], and has come to be regarded as one of the standard objects in the field. In recent years it has also played a role in areas as diverse as lowdimensional topology and geometric group theory [9], [12], [13], [31], [32] as well as the noncommutative version of probability [2], [3], [35], [41], [42], [43], [49], [50]. Due no doubt to its recent vintage, it is less well-known to the mathematical community at large than perhaps it deserves to be, but hopefully this short paper will help to remedy this state of affairs.…”

Section: Introductionmentioning

confidence: 99%

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Noncrossing Partitions in Surprising Locations

McCammond

2006

The American Mathematical Monthly

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Section: Braid Groupsmentioning

confidence: 98%

Section: Symmetric Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Noncrossing Partitions in Surprising Locations

McCammond

2006

The American Mathematical Monthly

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“…Moreover, Mladen Bestvina has used a weak version of non-positive curvature to show that finite-type Artin groups have essentially all of the expected group-theoretic consequences of non-positive curvature [8]. Finally, there is a finite Eilenberg-Maclane space for the braid groups constructed independently by Tom Brady [17] and Daan Krammer [60,61] and generalized to arbitrary Artin groups of finite-type by David Bessis [7] and by Tom Brady and Colum Watt [20]. If a piecewise Euclidean metric is assigned to one of these Brady-Krammer complexes then the corresponding Artin group will act geometrically on its universal cover.…”

Section: Definition 81 (Artin Groups and Coxeter Groups)mentioning

confidence: 99%

Constructing non-positively curved spaces and groups

McCammond¹

2009

Geometric and Cohomological Methods in Group Theory

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Abstract. The theory of non-positively curved spaces and groups is tremendously powerful and has enormous consequences for the groups and spaces involved. Nevertheless, our ability to construct examples to which the theory can be applied has been severely limited by an inability to test -in real timewhether a random finite piecewise Euclidean complex is non-positively curved. In this article I focus on the question of how to construct examples of nonpositively curved spaces and groups, highlighting in particular the boundary between what is currently do-able and what is not yet feasible. Since this is intended primarily as a survey, the key ideas are merely sketched with references pointing the interested reader to the original articles.Over the past decade or so, the consequences of non-positive curvature for geometric group theorists have been throughly investigated, most prominently in the book by Bridson and Haefliger [26]. See also the recent review article by Kleiner in the Bulletin of the AMS [59] and the related books by Ballmann [4], BallmannGromov-Schroeder [5] and the original long article by Gromov [48]. In this article I focus not on the consequences of the theory, but rather on the question of how to construct examples to which it applies. The structure of the article roughly follows the structure of the lectures I gave during the Durham symposium with the four parts corresponding to the four talks. Part 1 introduces the key problems and presents some basic decidability results, Part 2 focuses on practical algorithms and low-dimensional complexes, Part 3 presents case studies involving special classes of groups such as Artin groups, small cancellation groups and ample-twisted face pairing 3-manifolds. In Part 4 I explore the weaker notion of conformal non-positive curvature and introduce the notion of an angled n-complex. As will become clear, the topics covered have a definite bias towards research in which I have played some role. These are naturally the results with which I am most familiar and I hope the reader will pardon this lack of a more impartial perspective. Part 1. Negative curvatureAlthough the definitions of δ-hyperbolic and CAT(0) spaces / groups are wellknown, it is perhaps under-appreciated that there are at least seven potentially distinct classes of groups which all have some claim to the description "negativelycurved." Sections 1 and 2 briefly review the definitions needed in order to describe these classes, the known relationships between them, and the current status of the (rather optimisitic) conjecture that all seven of these classes are identical. Since the emphasis in this article is on the construction of examples, finite piecewise

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“…. , n) (see [3], [5]). In 1997, V. Reiner [11] introduced the lattice N C (B) (n) of non-crossing partitions of type B.…”

Section: Introductionmentioning

confidence: 99%

Enumerative Properties of NC (B) (p,q)

2011

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We determine the rank generating function, the zeta polynomial and the Möbius function for the poset N C (B) (p, q) of annular non-crossing partitions of type B, where p and q are two positive integers. We give an alternative treatment of some of these results in the case q = 1, for which this poset is a lattice. We also consider the general case of multiannular non-crossing partitions of type B, and prove that this reduces to the cases of non-crossing partitions of type B in the annulus and the disc.

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A Partial Order on the Symmetric Group and New K(π, 1)'s for the Braid Groups

Cited by 75 publications

References 9 publications

Noncrossing Partitions in Surprising Locations

Noncrossing Partitions in Surprising Locations

Constructing non-positively curved spaces and groups

Enumerative Properties of NC (B) (p,q)

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