Narrow Results

For any n>1, we construct examples of branched Galois coverings M→ℙⁿ where M is one of (ℙ¹)ⁿ, and , where is the 1-ball. In terms of orbifolds, this amounts to giving examples of orbifolds over ℙⁿ uniformized by M. We also discuss the related "orbifold braid groups".

Let Σ_g be a closed orientable surface of genus g and let Diff₀(Σ_g, area) be the identity component of the group of area-preserving diffeomorphisms of Σ_g. In this paper, we present the extension of Gambaudo–Ghys construction to the case of a closed hyperbolic surface Σ_g, i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of Σ_g defines a nontrivial homogeneous quasi-morphism on the group Diff₀(Σ_g, area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff₀(Σ_g, area) is infinite-dimensional. Let Ham(Σ_g) be the group of Hamiltonian diffeomorphisms of Σ_g. As an application of the above construction we construct two injective homomorphisms Z^m → Ham(Σ_g), which are bi-Lipschitz with respect to the word metric on Z^m and the autonomous and fragmentation metrics on Ham(Σ_g). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham(Σ_g).

In this paper, we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang–Baxter and boundary Yang–Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.

The topological explanation of the origin of Laughlin correlations in 2D charged systems under strong magnetic fields is formulated. Formal, self-consistent mathematical model of originally identified cyclotron braid subgroups is given in order to fully describe fundamentals of fractional quantum Hall effect, retrieve Laughlin correlations and point physical conditions which stand behind mysterious composite fermion structure. The new complete implementation of composite fermion basing on the first principles, without involving any artificial constructions (with flux-tubes or vortices) supply an explanation of previous models of composite fermions. Presented approach can lead to some corrections of numerical results in energy minimizations made within the traditional formulation of composite fermion model. Authors also identify the relations of FQHE in cyclotron braid terms within newly developing area of topological insulators and optical lattices. The prerequisites needed for formation of the fractional state are identified beyond the traditionally assumed factors, like the flat band condition and the interaction presence. The role of high mobility of carriers is highlighted in agreement with the experimental observations. Description, in terms of cyclotron braid subgroups, of the nature of yet unexplained novel experiments in Hall 2D systems including graphene is provided as well.

In this paper we show that the braid groups B_n, n≤6, have proper torsion-free non-abelian quotients. We also show that for n≤6 the homotopy braid groups are torsion-free.

The author previously introduced groups BV and as braided versions of groups introduced by Richard J. Thompson, and as having connections to braided tensor categories. Here we show that BV and are finitely presented by exhibiting finite presentations for them. We show that these groups are "Artinifications" of the older Thompson groups V and by showing that presentations for V and can be obtained from presentations for BV and by adding relations that trivialize the squares of the generators of the latter groups.

In this paper we construct a gathering process by the means of which we obtain new normal forms in braid groups. The new normal forms generalize Artin–Markoff normal forms and possess an extremely natural geometric description. In the two last sections of the paper we discuss the implementation of the introduced gathering process and the questions that arose in our work. This discussion leads us to some interesting observations, in particular, we offer a method of generating a random braid.

We prove that a construction similar to that described by Dehornoy in the case of braids is possible for every Artin–Tits group, yielding a partial ordering. A necessary condition for this partial order to be linear is that the associated Coxeter graph consists only of disjoint lines. So, in particular, type D is dismissed.

We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is, the limits of geodesic rays. In the case of the dual generators, it turns out that all boundary points are Busemann points, but this is not true for the Artin generators. We also characterize the geodesics with respect to the dual generators, which allows us to calculate the associated geodesic growth series.

In this paper, we give a description of the generators of the prime level congruence subgroups of braid groups. Also, we give a new presentation of the symplectic group over a finite field, and we calculate the symmetric quotients of the prime level congruence subgroups of braid groups. Finally, we find a finite generating set for the level-3 congruence subgroup of the braid group on three strands.

The mod 4 braid group, ${{𝒵}}_n$, is defined to be the quotient of the braid group by the subgroup of the pure braid group generated by squares of all elements, $P{B}_n^2$. Kordek and Margalit proved ${{𝒵}}_n$ is an extension of the symmetric group by ${\mathbb{Z}}_2^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}$. For $n\ge 1$, we construct a 2-cocycle in the group cohomology of the symmetric group with twisted coefficients classifying ${{𝒵}}_n$. We show this cocycle is the mod 2 reduction of the 2-cocycle corresponding to the extension of the symmetric group by the abelianization of the pure braid group. We also construct the 2-cocycle corresponding to this second extension and show that it represents an order two element in the cohomology of the symmetric group. Furthermore, we give presentations for both extensions and a normal generating set for $P{B}_n^2$.

In this paper we exhibit a way of obtaining linear representations of the braid groups B_n over ℤ[t] by studying their action on the set of isotopy classes of sets of simple closed curves on a punctured disc. The cases n=3, 4, 5 are shown to be very different from the cases n>5. We show a connection between representations of B₃ and Pascal's triangle. We also show that there is a sequence of polynomials κ_i(t), i≥0, related to polynomials P_i(t) defined by V. F. R. Jones all of whose roots give values of t for which these representations are not faithful.

In this paper, for every positive integer m, we define a representation ξ_n,m of the n-strand braid group B_n over a free ℤB_n+m-module. It not only provides an approach to construct new representations of braid groups, but also gives a new perspective to the homological representations such as the Lawrence–Krammer representation.

We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case.

We present a seemingly new normal form for braids, where every braid is expressed using a word in a regular language on some simple alphabet of elementary braids. This normal form stems from analysing the geometric action of braid groups on curves in a punctured disk.

The genus of knots is one of the fundamental invariants and can be seen as a complexity of knots. In this paper, we give a lower bound of the knot genus using the Dehornoy floor, which is a measure of complexity of braids in terms of the Dehornoy ordering.

We study quasi-morphisms on the groups P_n of pure braids on n strings and on the group of compactly supported area-preserving diffeomorphisms of an open two-dimensional disk. We show that it is possible to build quasi-morphisms on P_n by using knot invariants which satisfy some special properties. In particular, we study quasi-morphisms which come from knot Floer homology and Khovanov-type homology. We then discuss possible variations of the Gambaudo–Ghys construction, using the above quasi-morphisms on P_n to build quasi-morphisms on the group of diffeomorphisms of a 2-disk.

Computation of fundamental groups of Galois covers recently led to the construction and analysis of Coxeter covers of the symmetric groups [L. H. Rowen, M. Teicher and U. Vishne, Coxeter covers of the symmetric groups, J. Group Theory8 (2005) 139–169]. In this paper we consider analog covers of Artin's braid groups, and completely describe the induced geometric extensions of the braid group.

In this paper we constructed an isomorphic group of binary matrices to a finite symmetric group. Our method is based on the inversion of permutations. Using this embedding we find an algorithm for writing down a standard braid word representation for each positive permutation braid. Also an algorithm for writing basis of Hecke algebra H_n+1 from such basis of H_n is given.

In this note, we show that a homomorphism from the braid group B_n into an arbitrary group is injective if and only if its image can be left ordered in a particular way. It follows from this criterion that the image of B_n (n ≥ 2) under any homomorphism is bi-orderable if and only if it is infinite cyclic. As a second application, this criterion also provides a new proof that the Artin representation of B_n into Aut(F_n) is injective.