On braid groups

A dissertation in the Department of Mathematics submitted to the Faculty of the Graduate School of Arts and Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at New York University.     

On braid groups and right-angled Artin groups

In this article we prove a special case of a conjecture of A. Abrams and R. Ghrist about fundamental groups of certain aspherical spaces. Specifically, we show that the \(n\) -point braid group of a linear tree is a right-angled Artin group for each \(n\) .     

Twisted conjugacy in braid groups

In this note we solve the twisted conjugacy problem for braid groups, i.e., we propose an algorithm which, given two braids u, υ ∈ B _n and an automorphism φ ∈ Aut(B _n ), decides whether υ = (φ(x))^?1 ux for some x ∈ B _n . As a corollary, we deduce that each group of the form B _n ? H, a semidirect product of the braid group B _n by a torsion-free hyperbolic group H, has solvable conjugacy problem.     

Embedding right-angled Artin groups into graph braid groups

We construct an embedding of any right-angled Artin group G(Δ) defined by a graph Δ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of Δ. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.      

Transformation formulas for pseudo-characters of braid groups

We develop the theory of pseudo-characters of braid groups. We study a special family of operators between spaces of pseudo-characters of braid groups and describe techniques for obtaining new pseudo-characters of braid groups from known ones. A number of general results on the structure of the space of pseudo-characters is obtained. Bibliography: 12 titles.     

Cohomology of braid groups with nontrivial coefficients

In the paper, the homology of the braid groups with twisted coefficients and the homology of commutator subgroups of the braid groups are calculated. The main tool is the multiplicative structure on the homology induced by the “addition” of braid groups. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 846–854, June, 1996. This research was partially supported by the International Science Foundation under grant MQO000.     

Zariski theorems and diagrams for braid groups

Empirical properties of generating systems for complex reflection groups and their braid groups have been observed by Orlik-Solomon and Broué-Malle-Rouquier, using Shephard-Todd classification. We give a general existence result for presentations of braid groups, which partially explains and generalizes the known empirical properties. Our approach is invariant-theoretic and does not use the classification. The two ingredients are Springer theory of regular elements and a Zariski-like theorem. Oblatum 7-XII-2000 & 22-III-2001?Published online: 20 July 2001     

On the cycling operation in braid groups

The cycling operation is a special kind of conjugation that can be applied to elements in Artin’s braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In their seminal paper on braid-cryptography, Ko, Lee et al. proposed the cycling problem as a hard problem in braid groups that could be interesting for cryptography. In this paper we give a polynomial solution to that problem, mainly by showing that cycling is surjective, and using a result by Maffre which shows that pre-images under cycling can be computed fast. This result also holds in every Artin-Tits group of spherical type, endowed with the Artin Garside structure.On the other hand, the conjugacy search problem in braid groups is usually solved by computing some finite sets called (left) ultra summit sets (left-USSs), using left normal forms of braids. But one can equally use right normal forms and compute right-USSs. Hard instances of the conjugacy search problem correspond to elements having big (left and right) USSs. One may think that even if some element has a big left-USS, it could possibly have a small right-USS. We show that this is not the case in the important particular case of rigid braids. More precisely, we show that the left-USS and the right-USS of a given rigid braid determine isomorphic graphs, with the arrows reversed, the isomorphism being defined using iterated cycling. We conjecture that the same is true for every element, not necessarily rigid, in braid groups and Artin-Tits groups of spherical type.     

On the structure of surface pure braid groups

We describe some of the properties of the pure braid groups of surfaces different from and . In the case of compact, connected, orientable surfaces without boundary and of genus at least two, we give a necessary and sufficient condition for the splitting of the pure braid group exact sequence of Fadell and Neuwirth, thus answering completely a question of Birman.     

Representations of small braid groups in Temperley-Lieb algebra

In this paper we consider the question of faithfulness of the Jones' representation of braid group Bn into the Temperley-Lieb algebra TLn. The obvious motivation to study this problem is that any non-trivial element in the kernel of this representation (for any n) would almost certainly yield a non-trivial knot with trivial Jones polynomial (see [S. Bigelow, Does the Jones polynomial detect the unknot? J. Knot Theory Ramifications 11 (4) (2002) 493-505], we will explain it in more detail in Section 1). As one of the two main results we prove Theorem 1 in which we present a method to obtain non-trivial elements in the kernel of the representation of B₆ into TL9,2—to the authors' knowledge the first such examples in the second gradation of the Temperley-Lieb algebra. Theorem 2 which is a refinement of Theorem 1 may be used to produce smaller examples of the same kind. We also show briefly how some braids that are used in Section 4 to construct specific examples were generated with a computer program.     

On the structure of surface pure braid groups

We describe some of the properties of the pure braid groups of surfaces different from and . In the case of compact, connected, orientable surfaces without boundary and of genus at least two, we give a necessary and sufficient condition for the splitting of the pure braid group exact sequence of Fadell and Neuwirth, thus answering completely a question of Birman.     

Finite Hurwitz braid group actions for Artin groups

We prove that (with two possible exceptions) the Hurwitz braid group action on the sequence of standard generators of an irreducible Artin group has a finite orbit if and only if the Artin group is of finite type (i.e., the corresponding Coxeter group is finite).     

Local group representations

As a generalization of the usual notion of a group representation, a local group representation of connected and locally connected groups G is introduced as a certain sheaf A with base G, such that all fibres A _x may be identified with a fixed linear subspace D of the representation space. The sheaf provides a means of expressing a group symmetry which is known to hold only locally (in a sense which will be made precise). The problem of extending a local representation to a global one may be considered as a noncommutative counterpart of the dilation theory of systems of doubly commuting contractions. If G is a Lie group, one may associate a derived representation of the enveloping algebra (g) of the Lie algebra g of G. In case of isometric local representations in a Hilbert space, these are *-representations and determine the appropriate local representation uniquely. As an application we obtain that the n-tuples of antisymmetric operators of partial derivatives on L ²(), where is a connected open subset of ⁿ, determine up to translation.