Pure Braid Group Research Articles

A geometric interpretation of the normal closure of the braid group Bn in the braid group of the torus Bn(T)

It had been proved by Birman and Goldberg that the normal closure of the pure braid group [Formula: see text] in the pure braid group of the torus [Formula: see text] is the commutator subgroup [Formula: see text]. In this paper, we are going to study the case of full braid groups: i.e. the normal closure of [Formula: see text] in [Formula: see text], which turns out to have an interesting geometric description.

Word problem for special braid groups

In this paper, the main idea is to present the solvability of the word problem of pure virtual braid groups by using Gröbner-Shirshov basis theory.

Twists and braids for general 3-fold flops

Given a quasi-projective 3-fold X with only Gorenstein terminal singularities, we prove that the flop functors beginning at X satisfy higher degree braid relations, with the combinatorics controlled by a real hyperplane arrangement \mathcal H . This leads to a general theory, incorporating known special cases with degree 3 braid relations, in which we show that higher degree relations can occur even for two smooth rational curves meeting at a point. This theory yields an action of the fundamental group of the complexified complement \pi_1(\mathbb{C}^n\backslash \mathcal H_\mathbb{C}) on the derived category of X , for any such 3-fold that admits individually floppable curves. We also construct such an action in the more general case where individual curves may flop analytically, but not algebraically, and furthermore we lift the action to a form of affine pure braid group under the additional assumption that X is \mathbb Q -factorial. Along the way, we produce two new types of derived autoequivalences. One uses commutative deformations of the scheme-theoretic fibre of a flopping contraction, and the other uses noncommutative deformations of the fibre with reduced scheme structure, generalising constructions of Toda and the authors [T07, DW1] which considered only the case when the flopping locus is irreducible. For type A flops of irreducible curves, we show that the two autoequivalences are related, but in other cases they are very different, with the noncommutative twist being linked to birational geometry via the Bridgeland–Chen [B02, C02] flop-flop functor.

Almost-crystallographic groups as quotients of Artin braid groups

Let n,k≥3. In this paper, we analyse the quotient group Bn/Γk(Pn) of the Artin braid group Bn by the subgroup Γk(Pn) belonging to the lower central series of the Artin pure braid group Pn. We prove that it is an almost-crystallographic group. We then focus more specifically on the case k=3. If n≥5, and if τ∈N is such that gcd⁡(τ,6)=1, we show that Bn/Γ3(Pn) possesses an element of order τ if and only if Sn does, and we prove that there is a one-to-one correspondence between the conjugacy classes of elements of order τ in Bn/Γ3(Pn) with those of elements of order τ in the symmetric group Sn. We also exhibit a presentation for the almost-crystallographic group Bn/Γ3(Pn). Finally, we obtain some 4-dimensional almost-Bieberbach subgroups of B3/Γ3(P3), we explain how to construct almost-Bieberbach subgroups of B4/Γ3(P4) and B3/Γ4(P3), and we exhibit explicit elements of order 5 in B5/Γ3(P5).

Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph

Abstract In this paper, we show that the minimal asymptotic translation length of the Torelli group ${\mathcal{I}}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group ${\textrm{Mod}}(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translation length of the pure braid group ${\textrm{PB}}_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group ${\textrm{B}}_n$, which behaves like $1/n^2$.

On the Irreducibility of Fourth Dimensional Tuba's Representation of the Pure Braid Group on Three Strands

We consider Tuba's representation of the pure braid group, $%P_{3} $, given by the map $\phi :P_{3}\longrightarrow GL(4,F)$, where $F$ is an algebraically closed field. After, specializing the indeterminates used in defining the representation to non- zero complex numbers, we find sufficient conditions that guarantee the irreducibility of Tuba's representation of the pure braid group $P_{3}$ with dimension $d=4$. Under further restriction for the complex specialization of the indeterminates, we get a necessary and sufficient condition for the irreducibility of $\phi

Classifying Spaces for the Family of Virtually Cyclic Subgroups of Braid Groups

Abstract We prove that, for n ≥ 3, the minimal dimension of a model of the classifying space of the braid group $B_{n}$, and of the pure braid group $P_{n}$, with respect to the family of virtually cyclic groups is n.

Thompson groups for systems of groups, and their finiteness properties

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F , V , V_{\mathrm{br}} and F_{\mathrm{br}} arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pure braid groups. We give new examples of families of groups that admit a cloning system and study how the finiteness properties of the resulting generalized Thompson group depend on those of the original groups. The main new examples here include upper triangular matrix groups, mock reflection groups, and loop braid groups. For generalized Thompson groups of upper triangular matrix groups over rings of S -integers of global function fields, we develop new methods for (dis-)proving finiteness properties, and show that the finiteness length of the generalized Thompson group is exactly the limit inferior of the finiteness lengths of the groups in the family.

The Groups G2 n with Additional Structures

In the paper [1], V. O. Manturov introduced the groups G k n depending on two natural parameters n > k and naturally related to topology and to the theory of dynamical systems. The group G2 n , which is the simplest part of G k n , is isomorphic to the group of pure free braids on n strands. In the present paper, we study the groups G2 n supplied with additional structures–parity and points; these groups are denoted by G2n,p and G2n,d. First,we define the groups G2n,p and G2n,d, then study the relationship between the groups G2 n , G2n,p, and G2n,d. Finally, we give an example of a braid on n + 1 strands, which is not the trivial braid on n + 1 strands, by using a braid on n strands with parity. After that, the author discusses links in S g × S1 that can determine diagrams with points; these points correspond to the factor S1 in the product S g × S1.

Constructing real rational knots by gluing

We show that the problem of constructing an affine real rational knot of a reasonably low degree can be reduced to an algebraic problem involving the pure braid group: expressing an associated element of the pure braid group in terms of the standard generators of the pure braid group. We also predict the existence of a real rational knot in a degree that is expressed in terms of the edge number of its polygonal representation.

Formal descriptions of Turaev's loop operations

Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra \mathbb Q[\pi] of the fundamental group \pi of a surface with boundary. The first operation is binary and measures the intersection of two oriented based curves on the surface, while the second operation is unary and computes the self-intersection of an oriented based curve. It is already known that Turaev's intersection pairing has an algebraic description when the group algebra \mathbb Q[\pi] is completed with respect to powers of its augmentation ideal and is appropriately identified to the degree-completion of the tensor algebra T(H) of H:=H_1(\pi;\mathbb Q) . In this paper, we obtain a similar algebraic description for Turaev's self-intersection map in the case of a disk with p punctures. Here we consider the identification between the completions of \mathbb Q[\pi] and T(H) that arises from a Drinfeld associator by embedding \pi into the pure braid group on (p+1) strands; our algebraic description involves a formal power series which is explicitly determined by the associator. The proof is based on some three-dimensional formulas for Turaev's loop operations, which involve 2 -strand pure braids and are shown for any surface with boundary.

Group orderings, dynamics, and rigidity

Let G be a countable group. We show there is a topological relationship between the space CO(G) of circular orders on G and the moduli space of actions of G on the circle; as well as an analogous relationship for spaces of left orders and actions on the line. In particular, we give a complete characterization of isolated left and circular orders in terms of strong rigidity of their induced actions of G on $S^1$ and R. As an application of our techniques, we give an explicit construction of infinitely many nonconjugate isolated points in the spaces CO(F_{2n}) of circular orders on free groups disproving a conjecture from Baik--Samperton, and infinitely many nonconjugate isolated points in the space of left orders on the pure braid group P_3, answering a question of Navas. We also give a detailed analysis of circular orders on free groups, characterizing isolated orders.

Finite orbits of the pure braid group on the monodromy of the 2-variable Garnier system

In this paper we show that the $SL_{2}(\mathbb C)$ character variety of the Riemann sphere $\Sigma_5$ with five boundary components is a $5$-parameter family of affine varieties of dimension $4$. We endow this family of affine varieties with an action of the braid group and classify exceptional finite orbits. This action represents the nonlinear monodromy of the $2$ variable Garnier system and finite orbits correspond to algebraic solutions.

A Tannakian Interpretation of the Elliptic Infinitesimal Braid Lie Algebras

Let n ≥ 1. The pro-unipotent completion of the pure braid group of n points on a genus 1 surface has been shown to be isomorphic to an explicit pro-unipotent group with graded Lie algebra using two types of tools: (a) minimal models (Bezrukavnikov), (b) the choice of a complex structure on the genus 1 surface, making it into an elliptic curve E, and an appropriate flat connection on the configuration space of n points in E (joint work of the authors with D. Calaque). Following a suggestion by P. Deligne, we give an interpretation of this isomorphism in the framework of the Riemann-Hilbert correspondence, using the total space E # of an affine line bundle over E, which identifies with the moduli space of line bundles over E equipped with a flat connection.

Arithmetic Topology in Ihara Theory

Ihara initiated the study of a certain Galois representation that may be seen as an arithmetic analogue of the Artin representation of a pure braid group. We pursue the analogies in Ihara theory further and give foundational results, following some issues and their interrelations in the theory of braids and links such as Milnor invariants, Johnson homomorphisms, Magnus–Gassner cocycles and Alexander invariants, and study relations with arithmetic in Ihara theory.

Separation in the BNSR-invariants of the pure braid groups

We inspect the BNSR-invariants Σm(Pn) of the pure braid groups Pn, using Morse theory. The BNS-invariants Σ1 (Pn) were previously computed by Koban, McCammond, and Meier. We prove that for any 3≤m≤n, the inclusion Σm−2 (Pn) ⊆ Σm−3 (Pn) is proper, but Σ∞(Pn) = Σn−2 (Pn). We write down explicit character classes in each relevant Σm−3 (Pn)\Σm−2 (Pn). In particular we get examples of normalsubgroups N ≤ Pn with Pn/N ∼= Z such that N is of type Fm−3 but not Fm−2, for all 3 ≤ m ≤ n.

Automorphisms of pure braid groups

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to an automorphism of $B_n$.

Subgroup graph methods for presentations of finitely generated groups and the contractibility of associated simplicial complexes

In this article we generalize the theory of subgroup graphs of subgroups of free groups, developed by I. Kapovich and A. Myasnikov, based on a work by J. Stallings, to finite index subgroups of finitely generated groups. Given a presentation of a finitely generated group G=〈X|R〉 and a finite connected X-regular graph Γ which fulfills the relators R, we associate to Γ a finite index subgroup H of G. Conversely, the Schreier coset graph of H with respect to X and G is such a graph Γ. Firstly, we study and prove various properties of H in relation to the graph Γ. Secondly, we prove that for many finitely generated infinite groups the order and nerve complexes that we associate to G are contractible. In particular, this is the case for free and free abelian groups, Fuchsian groups of genus g≥2, infinite right angled Coxeter groups, Artin and pure braid groups, infinite virtually cyclic groups, Baumslag–Solitar groups as well as the (free) product of at least two of these, and all finite index subgroups of these groups.

Polynomial Splitting Measures and Cohomology of the Pure Braid Group

We study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\mathbb{F}_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $\lambda$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \mathbb{Q})$. We deduce that the splitting measures for all parameter values $z= -\frac{1}{m}$ (resp. $z= \frac{1}{m}$), after rescaling, are characters of $S_n$-representations (resp. virtual $S_n$-representations.)

On the Center-Valued Atiyah Conjecture for $L^2$-Betti Numbers

The so-called Atiyah conjecture states that the \mathcal N(G) -dimensions of the L^2 -homology modules of finite free G -CW-complexes belong to a certain set of rational numbers, depending on the finite subgroups of G . In this article we extend this conjecture to a statement for the center-valued dimensions. We show that the conjecture is equivalent to a precise description of the structure as a semisimple Artinian ring of the division closure D(\Bbb{Q}[G]) of \Bbb{Q}[G] in the ring of affiliated operators. We prove the conjecture for all groups in Linnell's class \frak{C} , containing in particular free-by-elementary amenable groups. The center-valued Atiyah conjecture states that the center-valued L^2 -Betti numbers of finite free G -CW-complexes are contained in a certain discrete subset of the center of \Bbb{C}[G] , the one generated as an additive group by the center-valued traces of all projections in \Bbb{C}[H] , where H runs through the finite subgroups of G . Finally, we use the approximation theorem of Knebusch [15] for the center-valued L^2 -Betti numbers to extend the result to many groups which are residually in \frak{C} , in particular for finite extensions of products of free groups and of pure braid groups.