Surface Braid Research Articles

From pure braid groups to hyperbolic groups

In this note we show that any homomorphism from a pure surface braid group to a torsion-free hyperbolic group either has a cyclic image or factors through a forgetful map. This extends and gives a new proof of an earlier (2019) result of the author which works only when the target is a free group or a surface group. We also prove a similar rigidity result for the pure braid group of the disk.

Distinguishing surface-links described by 4-charts with two crossings and eight black vertices

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface (called a surface-link) embedded in 4-space. In this paper, we investigate surface-links by using charts. In [T. Nagase and A. Shima, The structure of a minimal [Formula: see text]-chart with two crossings I: Complementary domains of [Formula: see text], J. Knot Theory Ramifactions 27(14) (2018) 1850078; T. Nagase and A. Shima, The structure of a minimal [Formula: see text]-chart with two crossings II: Neighbourhoods of [Formula: see text], Revista de la Real Academia de Ciencias Exactas, Fiskcas y Natrales. Serie A. Math. 113 (2019) 1693–1738, arXiv:1709.08827v2 ] we gave an enumeration of the charts with two crossings. In particular, there are two classes for 4-charts with two crossings and eight black vertices. The first class represents surface-links each of which is connected. The second class represents surface-links each of which is exactly two connected components. In this paper, by using quandle colorings, we shall show that the charts in the second class represent different surface-links.

Properties of minimal charts and their applications IX: charts of type [formula omitted

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let Γ be a chart, and we denote by Γm the union of all the edges of label m. A chart Γ is of type (4,3) if there exists a label m such that w(Γ)=7, w(Γm∩Γm+1)=4, w(Γm+1∩Γm+2)=3 where w(G) is the number of white vertices in G. In this paper, we prove that there is no minimal chart of type (4,3).

Extra-special quotients of surface braid groups and double Kodaira fibrations with small signature

We study some special systems of generators on finite groups, introduced in previous work by the first author and called diagonal double Kodaira structures, in order to investigate finite non-abelian quotients of the pure braid group on two strands {mathsf {P}}_2(Sigma _b), where Sigma _b is a closed Riemann surface of genus b. In particular, we prove that, if a finite group G admits a diagonal double Kodaira structure, then |G|ge 32, and equality holds if and only if G is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two 3-dimensional families of double Kodaira fibrations having signature 16. Such surfaces are different from the ones recently constructed by Lee, Lönne and Rollenske and, as far as we know, they provide the first examples of positive-dimensional families of double Kodaira fibrations with small signature.

Properties of minimal charts and their applications VIII: Charts of type (7)

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let [Formula: see text] be a chart, and we denote by [Formula: see text] the union of all the edges of label [Formula: see text]. A chart [Formula: see text] is of type [Formula: see text] if there exists a label [Formula: see text] such that [Formula: see text], [Formula: see text] where [Formula: see text] is the number of white vertices in [Formula: see text]. In this paper, we prove that there is no minimal chart of type [Formula: see text].

Surface braid groups, Finite Heisenberg covers and double Kodaira fibrations

We exhibit new examples of double Kodaira fibrations by using finite Galois covers of a product $\Sigma_b \times \Sigma_b$, where $\Sigma_b$ is a smooth projective curve of genus $b \geq 2$. Each cover is obtained by providing an explicit group epimorphism from the pure braid group $\mathsf{P}_2(\Sigma_b)$ to some finite Heisenberg group. In this way, we are able to show that every curve of genus $b$ is the base of a double Kodaira fibration; moreover, the number of pairwise non-isomorphic Kodaira fibred surfaces fibering over a fixed curve $\Sigma_b$ is at least $\boldsymbol{\omega}(b+1)$, where $\boldsymbol{\omega} \colon \mathbb{N} \to \mathbb{N}$ stands for the arithmetic function counting the number of distinct prime factors of a positive integer. As a particular case of our general construction, we obtain a real $4$-manifold of signature $144$ that can be realized as a real surface bundle over a surface of genus $2$, with fibre genus $325$, in two different ways. This provides (to our knowledge) the first "double" solution to a problem from Kirby's list in low-dimensional topology.

Lower central series, surface braid groups, surjections and permutations

AbstractGeneralising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ $\mathbb N$ for which there exists a surjection between the n- and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.

Crystallographic groups and flat manifolds from surface braid groups

Let M be a compact surface without boundary, and n≥2. We analyse the quotient group Bn(M)/Γ2(Pn(M)) of the surface braid group Bn(M) by the commutator subgroup Γ2(Pn(M)) of the pure braid group Pn(M). If M is different from the 2-sphere S2, we prove that Bn(M)/Γ2(Pn(M))≅Pn(M)/Γ2(Pn(M))⋊φSn, and that Bn(M)/Γ2(Pn(M)) is a crystallographic group if and only if M is orientable.If M is orientable, we prove a number of results regarding the structure of Bn(M)/Γ2(Pn(M)). We characterise the finite-order elements of this group, and we determine the conjugacy classes of these elements. We also show that there is a single conjugacy class of finite subgroups of Bn(M)/Γ2(Pn(M)) isomorphic either to Sn or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection Bn(M)/Γ2(Pn(M))⟶Sn is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups G˜n,g of Bn(M)/Γ2(Pn(M)) of dimension 2ng and whose holonomy group is the finite cyclic group of order n, and if Xn,g is a flat manifold whose fundamental group is G˜n,g, we prove that it is an orientable Kähler manifold that admits Anosov diffeomorphisms.

ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

AbstractThe closure of a braid in a closed orientable surface Ʃ is a link in Ʃ ×S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

Lower central and derived series of semi-direct products, and applications to surface braid groups

For an arbitrary semi-direct product, we give a general description of its lower central series and an estimation of its derived series. In the second part of the paper, we study these series for the full braid group Bn(M) and pure braid group Pn(M) of a compact surface M, orientable or non-orientable, the aim being to determine the values of n for which Bn(M) and Pn(M) are residually nilpotent or residually soluble. We first solve this problem in the case where M is the 2-torus. We then use the results of the first part of the paper to calculate explicitly the lower central series of Pn(K), where K is the Klein bottle. Finally, if M is a non-orientable, compact surface without boundary, we determine the values of n for which Bn(M) is residually nilpotent or residually soluble in the cases that were not already known in the literature.

Surjective homomorphisms between surface braid groups

Let $PB_n(S_{g,p})$ be the pure braid group of a genus $g>1$ surface with $p$ punctures. In this paper we prove that any surjective homomorphism $PB_n(S_{g,p})\to PB_m(S_{g,p})$ factors through one of the forgetful homomorphisms. We then compute the automorphism group of $PB_n(S_{g,p})$, extending Irmak, Ivanov and McCarthy's result to the punctured case. Surprisingly, in contrast to the $n=1$ case, any automorphism of $PB_n(S_{g,p})$, $n>1$ is geometric.

On the non-realizability of braid groups by diffeomorphisms

For every compact surface $S$ of finite type (possibly with boundary components but without punctures), we show that when $n$ is sufficiently large there is no lift $\sigma$ of the surface braid group $B_n(S)$ to $\operatorname{Diff}(S,n)$, the group of $C^1$ diffeomorphisms preserving $n$ marked points and restricting to the identity on the boundary. Our methods are applied to give a new proof of Morita's non-lifting theorem in the best possible range. These techniques extend to the more general setting of spaces of codimension-$2$ embeddings, and we obtain corresponding results for spherical motion groups, including the string motion group.

OEandW* superrigidity results for actions by surface braid groups

We show that several important normal subgroups $\Gamma$ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure preserving action $\Gamma \curvearrowright X$ is stably OE-superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings. Using these results in combination with previous results from [CIK13] we deduce that any free, ergodic, probability measure preserving action of almost any surface braid group is stably W*-superrigid, i.e., it can be completely reconstructed from its von Neumann algebra.

Primeness Results for von Neumann Algebras Associated with Surface Braid Groups

In this paper we introduce a new class of non-amenable groups denoted by ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ which give rise to $\textit{prime}$ von Neumann algebras. This means that for every $\Gamma \in {\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ its group von Neumann algebra $L(\Gamma)$ cannot be decomposed as a tensor product of diffuse von Neumann algebras. We show ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ is fairly large as it contains many examples of groups intensively studied in various areas of mathematics, notably: all infinite central quotients of pure surface braid groups; all mapping class groups of (punctured) surfaces of genus $0,1,2$; most Torelli groups and Johnson kernels of (punctured) surfaces of genus $0,1,2$; and, all groups hyperbolic relative to finite families of residually finite, exact, infinite, proper subgroups.

On the geometry of some unitary Riemann surface braid group representations and Laughlin-type wave functions

In this note we construct the simplest unitary Riemann surface braid group representations geometrically by means of stable holomorphic vector bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail also with the help of noncommutative geometric tools, and an application of Fourier–Mukai–Nahm techniques is also given, explaining the emergence of an intriguing Riemann surface braid group duality.

On Cohen braids

For a general connected surface M and an arbitrary braid α from the surface braid group Bn−1(M), we study the system of equations d1β = … = dnβ = α, where the operation di is the removal of the ith strand. We prove that for M ≠ S2 and M ≠ ℝP2, this system of equations has a solution β ∈ Bn(M) if and only if d1α = … = dn−1α. We call the set of braids satisfying the last system of equations Cohen braids. We study Cohen braids and prove that they form a subgroup. We also construct a set of generators for the group of Cohen braids. In the cases of the sphere and the projective plane we give some examples for a small number of strands.

Satellites of an oriented surface link and their local moves

For an oriented surface link F in R4, we consider a satellite construction of a surface link, called a 2-dimensional braid over F, which is in the form of a covering over F. We introduce the notion of an m-chart on a surface diagram π(F)⊂R3 of F, which is a finite graph on π(F) satisfying certain conditions and is an extended notion of an m-chart on a 2-disk presenting a surface braid. A 2-dimensional braid over F is presented by an m-chart on π(F). It is known that two surface links are equivalent if and only if their surface diagrams are related by a finite sequence of ambient isotopies of R3 and local moves called Roseman moves. We show that Roseman moves for surface diagrams with m-charts can be well-defined.

On charts with two crossings II

Let $\Gamma$ be a chart with at most two crossings. In this paper, we show that if $\Gamma$ is a 2-minimal generalized $n$-chart with $n \ge 5$, then $\Gamma$ contains at least $4n-10$ black vertices. And we show that if the closure of the surface braid represented by $\Gamma$ is a disjoint union of spheres, then $\Gamma$ is a ribbon chart. Hence the closure is a ribbon surface.

A Birman exact sequence for [formula omitted

The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove that in the context of the automorphism group of a free group, the natural kernel is finitely generated. However, it is not finitely presentable; indeed, we prove that its second rational homology group has infinite rank by constructing an explicit infinite collection of linearly independent abelian cycles. We also determine the abelianization of our kernel and build a simple infinite presentation for it. The keys to many of our proofs are several new generalizations of the Johnson homomorphisms.

Predicting the elastic moduli of three-dimensional (four-step) braided tubes using a spatial spring model

In the past, the fiber bundles were mostly meshed for computing the mechanical properties of the unit cell in fabric composites. Consequently, not only the degrees of freedom but also the complexities in computation are increased. To improve the efficiency in analysis, in this study, a previously developed spring model was extended to predict the effective elastic moduli of three-dimensional (four-step) braided tubes. In the meanwhile, a related compression test was conducted to validate the computational results. It is shown that the results from the computation agree well with the experimental results. In addition, the effects of fabric parameters (such as surface braid angle and fiber volume fraction in a fan-shaped unit cell) on the effective elastic moduli of three-dimensional (four-step) braided tubes were also investigated.