The paper examines subgroup separability for surface and virtual braid groups.
problem Subgroup separability of surface and virtual braid groups.
method Study of subgroup separability (LERF) properties.
result Properties of subgroup separability for surface and virtual braid groups are explored.
In this paper we introduce the framed pure braid group on n strands of an oriented surface, a topological generalisation of the pure braid group Pn. We give different equivalents definitions for framed pure braid groups and we study exact sequences relating these groups with other generalisations of Pn, usually…
New representation for braid groups and surface braid groups, extending Lawrence-Krammer-Bigelow.
problem Constructing representations for braid groups and surface braid groups.
method Pro-nilpotent tower of representations, starting with the original LKB representation.
result 3-variable enrichment of the Lawrence-Krammer-Bigelow representation.
We give presentations of braid groups and pure braid groups on surfaces.
Paper studies R∞-property for braid groups on surfaces, proving it for most cases.
problem Investigating the R∞-property for braid groups on orientable surfaces.
method Analyzing surface pure and full braid groups to determine the R∞-property.
result Most surface braid groups have the R∞-property, with exceptions noted.
Article provides conditions for embedding braid groups into mapping class groups.
problem Embedding braid groups into mapping class groups.
method Necessary and sufficient condition for embedding finite index subgroups.
result Conditions for embedding braid groups into mapping class groups.
The paper studies Coxeter quotients of surface braid groups.
problem Analyzing Coxeter quotients of surface braid groups.
method Analyzing Coxeter-type quotients of surface braid groups by normal closure and commutator relations.
result Characterized Coxeter quotients of surface braid groups.
Corrects earlier work on surface orbifold pure braid groups.
problem Proving a four-term exact sequence for surface orbifold pure braid groups.
method Analyzes surface orbifold pure braid groups for all genus ≥ 1, 2D orientable orbifolds with cone points.
result Proves a four-term exact sequence for surface orbifold pure braid groups.
Characterizes multitwists in surface mapping class groups.
problem Understanding multitwists in surface mapping class groups.
method Characterization through braid relations.
result Characterized multitwists satisfying braid relations.
We determine the lower central series and corresponding residual properties for braid groups and pure braid groups of orientable surfaces.
Sharp bounds found on nonabelian quotients of surface braid groups.
problem Finding the smallest nonabelian quotients of surface braid groups.
method Sharp lower bounds and classification of quotients.
result Quotients of minimum order are either symmetric groups or 2-step nilpotent p-groups.
Study of decorated surfaces with vortices and their group structures.
problem Understanding group structures of decorated surfaces with vortices.
method Proved isomorphism between cluster braid group, braid twist group, and fundamental group of moduli space.
result Finite presentations of isomorphic groups were given.
New proof shows homomorphisms from pure braid groups to hyperbolic groups have cyclic images or factor through forgetful maps.
problem Characterizing homomorphisms from pure braid groups to hyperbolic groups.
method Extending and proving a new rigidity result for pure braid groups, focusing on homomorphisms to hyperbolic groups.
result Homomorphisms from pure braid groups to hyperbolic groups either have cyclic images or factor through a forgetful map.
The closure of a braid in a closed orientable surface Σ is a link in Σ×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 t…
Classifies homomorphisms from braid groups to mapping class groups of nonorientable surfaces.
problem Classifying homomorphisms from braid groups to mapping class groups of nonorientable surfaces.
method Classifies homomorphisms based on the properties of Dehn twists and crosscap transpositions.
result Every homomorphism is either cyclic or maps generators to distinct Dehn twists or crosscap transpositions.
We prove that the pure braid groups on closed, orientable surfaces are bi-orderable, and that the pure braid groups on closed, non-orientable surfaces have generalized torsion, thus they are not bi-orderable.
In this paper, we prove that each automorphism of a surface braid group is induced by a homeomorphism of the underlying surface, provided that this surface is a closed, connected, orientable surface of genus at least 2, and the number of strings is at least three. This result generalizes previous results for classical …
The abstract discusses braided surfaces and their characteristic maps, linking them to algebraic and geometric properties.
problem Characterizing braided surfaces and their characteristic maps.
method Using branched coverings and factorization through free groups, the abstract explores the algebraic and geometric properties of braided surfaces.
result The abstract establishes a connection between braided surfaces and characteristic maps, providing new insights into their structure.
Polynomial representations found in surface braid and mapping class groups.
problem Homological representations of surface braid and mapping class groups.
method Study of homological representation functors and short exact sequences.
result Many homological representation functors are polynomial.
New groups from surface braids help create complex geometric shapes.
problem Creating new geometric shapes from surface braids.
method Using finite quotients of surface braid groups to construct double Kodaira fibrations.
result New geometric shapes (double Kodaira fibrations) created using these groups.
In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.
Braid combing is a procedure defined by Emil Artin to solve the word problem in braid groups for the first time. It is well-known to have exponential complexity. In this paper, we use the theory of straight line programs to give a polynomial algorithm which performs braid combing. This procedure can be applied to braid…
This paper classifies all admissible quotients of a surface braid group up to order 127.
problem Classifying admissible quotients of surface braid groups with bounded order.
method Analyzing finite quotients of the pure braid group on two strands of a Riemann surface.
result All admissible quotients of the braid group have order at most 127.
We give a survey of the theory of surface braid groups and the lower algebraic K-theory of their group rings. We recall several definitions and describe various properties of surface braid groups, such as the existence of torsion, orderability, linearity, and their relation both with mapping class groups and with the h…
We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.
Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ 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, …
New research shows all intermediate subgroups of braid groups are not bi-orderable.
problem Understanding bi-orderability of braid groups and their subgroups.
method Analyzing pure and full braid groups of various surfaces.
result Intermediate subgroups of braid groups are not bi-orderable.
The article calculates the minimal model dimensions for classifying spaces of surface braid groups.
problem Classifying spaces for families of virtually abelian subgroups of surface braid groups.
method Analyzes the pure and full braid groups of surfaces with at least one boundary component or one puncture, proving analogous results for amenable subgroups.
result Minimal dimensions of models for classifying spaces are equal to the virtual cohomological dimension plus the rank of virtually abelian subgroups.
For every group genetic code with finite number of generating and at most with one defining relation we introduce the braid group of this genetic code. This construction includes the braid group of Euclidean plane, the braid groups of closed orientable surfaces, B type groups of Artin-Brieskorn, and allow us to study a…
Study the relationship between orbit braid group and equivariant mapping class group on surfaces.
problem Understanding the relationship between mapping class groups and braid groups with group actions.
method Using the fibration F0GMightarrowF(M/G,n) and exact sequence. result The conclusion is closely connected with the braid group of the quotient space.
Study of polynomial strata using braid groups and translation surfaces.
problem Understanding the monodromy of polynomial strata.
method Using infinite-area translation surfaces and braid groups.
result Determine the monodromy of polynomial strata in the braid group.
New findings on algebraic structure of hyperbolic graph braid groups.
problem Classifying and understanding the algebraic structure of hyperbolic graph braid groups.
method Analyzing specific graph types (sun and pulsar graphs) and proving theorems about their braid groups.
result 3-strand braid groups of sun graphs are free, while most pulsar graphs contain surface subgroups.
The paper classifies homomorphisms between braid groups and mapping class groups.
problem Classifying homomorphisms between braid groups and mapping class groups.
method Analyzing specific cases and using surface bundles over configuration spaces.
result Sharp classification of homomorphisms, showing cyclic or standard representations.
We consider exact sequences and lower central series of surface braid groups and we explain how they can prove to be useful for obtaining representations for surface braid groups. In particular, using a completely algebraic framework, we describe the notion of extension of a representation introduced and studied recent…
The study examines when braid groups of manifolds are Kähler.
problem When are braid groups of manifolds Kähler?
method The approach uses previous results on Kähler groups and homological properties of braid groups.
result With some exceptions, the pure braid group of a Riemann surface with at least 2 strands is never Kähler.
The paper defines biquandles for groups and constructs a homomorphism.
problem Describing dynamical systems in configuration systems.
method Defined biquandles on groups Gnk and constructed a homomorphism. result Homomorphism from surface singular braid monoid to Gn2. Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give sever…
Study extends symmetries of sphere points to surface mapping classes.
problem Understanding symmetries of points on spheres and their connections.
method Establishing isomorphisms between moduli spaces and mapping class groups.
result Finitely many integral braid group orbits in rank 4 Stokes matrices.
We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit an universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the considered surface.
We define the notion of a braided link cobordism in S3×[0,1], which generalizes Viro's closed surface braids in R4. We prove that any properly embedded oriented surface W⊂S3×[0,1] is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary wh…
Study deformation spaces of irregular isomonodromy systems on Riemann surfaces.
problem Understanding the topology of irregular isomonodromy systems.
method Define and study moduli spaces of deformations of irregular classes on Riemann surfaces.
result Generalize G-braid groups to study fundamental groups of deformation spaces.
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 g…
Study algebraic K-theory for specific groups of non-orientable surfaces.
problem Algebraic K-theory of group rings for specific non-orientable surface groups.
method Detailed analysis of group rings and algebraic K-theory.
result General formula for algebraic K-theory groups of mapping class groups of non-orientable surfaces.
We prove that if g and n are integers at least two, then the abstract commensurator of the braid group with n strands on a closed orientable surface of genus g is naturally isomorphic to the extended mapping class group of a compact orientable surface of genus g with n boundary components.
The Dehornoy order on braid groups is derived from a cluster algebra.
problem Deriving the Dehornoy order on braid groups from a cluster algebra.
method Using a tracial state on a cluster C∗-algebra associated to a surface. result The Dehornoy order on B2g+n is recovered from the cluster algebra. Researchers compute spin structures on hyperelliptic curves using braid groups.
problem Understanding spin structures on hyperelliptic curves of genus g.
method Action of the Artin braid group B_{2g+2} on spin structures, combinatorial computation of orbits and isotropy groups.
result Purely combinatorial computation of S_{2g+2}-orbits and isotropy groups of spin structures.
Researchers map the fundamental group of polynomial strata to a braid group.
problem Understanding the fundamental group of polynomial strata.
method Analyzing the logarithmic derivative of polynomials to determine the map to a braid group.
result The map from the fundamental group of a stratum to a braid group is characterized by the geometry of the translation surface structure.
Study on when the lower central series stops for various groups, including braid groups.
problem Understanding when the lower central series stops for different groups.
method Various techniques applied to braid groups and related groups.
result Complete computation of the lower central series for most groups studied.