We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic -space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient. These generators have the most natural fo…
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48 results for “complex hyperbolic braid group”
New insights into a complex hyperbolic braid group quotient.
problem Understanding a complex hyperbolic braid group quotient.
method Analyzing the moduli space of 12-tuples in CP1 and identifying loops.
result Identifying loops in the 9-ball quotient corresponding to standard braid generators.
The paper defines parabolic subgroups for complex braid groups and proves they form a lattice.
problem Defining and characterizing parabolic subgroups in complex braid groups.
method Introducing and studying parabolic subgroups of generalized braid groups associated with complex reflection groups.
result Parabolic subgroups form a lattice in most cases, with specific properties and conjectures about hyperbolicity.
Study large-scale geometry of graph braid groups via cubical structures.
problem Classify and understand the quasi-isometry of graph braid groups.
method Exploit cubical structures to relate hyperbolicity, undistorted subgroups, and group decompositions.
result Complete classification of graph braid groups quasi-isometric to free groups.
We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from n dimensional complex vector space, or n dimensional complex hyperbolic space, or the Hermitian symmetric space for O(2,n), and then takes the quotient by a discrete …
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.
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.
Defines fundamental racks for braid spaces of complex reflection groups.
problem Understanding fundamental racks for braid spaces of complex reflection groups.
method Defines an augmented rack associated to the orbifold fundamental group.
result Yields representations of the orbifold fundamental group on the cohomology of the rack space.
Paper constructs representations for virtual braids and flat braids.
problem Calculating hyperbolic volumes of knot complements.
method Cluster algebra approach for virtual braid group.
result Forbidden relations do not hold in virtual braid group representation.
The paper classifies when certain graph braid groups are 3-manifold groups.
problem Identifying when graph braid groups are 3-manifold groups.
method Analyzing the graph braid groups for specific graphs .
result The paper shows that is a 3-manifold group, but is not quasi-isometric to a 3-manifold group for .
The paper finds Artin presentations for the trivial group and identifies hyperbolic 3-braids.
problem Artin presentations for the trivial group and hyperbolic 3-braids.
method Analysis of the hexatangle and Dehn surgery.
result Artin presentations for the trivial group and characterization of hyperbolic 3-braids.
Cusped spaces for hierarchically hyperbolic groups, and applications to Dehn filling quotientsmath.GT
Study of Dehn filling quotients in hierarchically hyperbolic groups.
problem Understanding the structure of Dehn filling quotients in specific groups.
method Introduced a construction for cusped spaces of relatively hyperbolic groups and used it to study Dehn-filling-like quotients.
result Infinite hyperbolic quotients of mapping class groups of punctured spheres and braid groups are found.
We construct an embedding of any right-angled Artin group defined by a graph into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of . This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid g…
Geometrically describes hyperbolic structures on link complements using quantum groups.
problem Describing hyperbolic structures on link complements algebraically.
method Uses octahedral decomposition and Kashaev-Reshetikhin's braiding on quantum group .
result Shows how to interpret geometrically the algebraic gluing equations for hyperbolic structures.
We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the -norm metric); this extends resul…
We present and discuss some open problems formulated by participants of the International Workshop "Knots, Braids, and Auto\-mor\-phism Groups" held in Novosibirsk, 2014. Problems are related to palindromic and commutator widths of groups; properties of Brunnian braids and two-colored braids, corresponding to an amalga…
It is well-known that there is a faithful representation of braid groups on automorphism groups of free groups, and it is also well-known that free groups are bi-orderable. We investigate which n-strand braids give rise to automorphisms which preserve some bi-ordering of the free group rank n. As a consequence of our w…
We show that a non-trivial, non-central normal subgroup of the braid groups contains a braid whose closure is a hyperbolic knot with arbitrary large genus. This shows that non-faithfulness of a quantum representation implies that the corresponding quantum invariant fails to detect the unknot. The proof utilizes the Deh…
Completes results on complex braid group parabolic subgroups.
problem Proves properties of complex braid group parabolic subgroups.
method Uses Garside groupoid structure of B(G31) to extend results.
result Proves main theorems for B(G31) parabolic subgroups.
New groups connect braids and 3-manifolds.
problem Understanding relationships between braids and 3-manifolds.
method Introducing and studying groups .
result Groups lead to new relationships between braids and manifolds.
The braid group of a complex reflection group is shown to be an index d subgroup.
problem Understanding the structure of braid groups associated with complex reflection groups.
method Presented a compatible presentation for the braid group of the orbifold quotient and a tagged triangulation of the disk.
result The braid group of the complex reflection group is an index subgroup of the braid group of the orbifold quotient.
We investigate the average-case complexity of decision problems for finitely generated groups, in particular the word and membership problems. Using our recent results on ``generic-case complexity'' we show that if a finitely generated group has the word problem solvable in subexponential time and has a subgroup of…
We prove that an arbitrary right-angled Artin group admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk an…
New link groups are derived from torus necklaces, connecting braid groups to reflection groups.
problem Understanding the relationship between braid groups and reflection groups.
method Constructing torus necklaces and linking them to braid groups of -reflection groups.
result Link groups of torus necklaces are precisely braid groups of -reflection groups, with meridians as braid reflections.
We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperb…
Study on embedding tree products into groups, distinguishing them.
problem Quasi-isometric embedding of tree products into various groups.
method Using coarse embeddings of products of bushy trees into hierarchically hyperbolic spaces.
result Quasi-isometrically distinguish and rule out embeddings between groups.
Classifies certain graph 2-braid groups up to quasi-isometry.
problem Classifying 2-braid groups over graphs up to quasi-isometry.
method Using intersection complexes and right-angled Artin groups.
result Classifies 2-braid groups over graphs with circumference ≤ 1 up to quasi-isometry.
Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids o…
The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natu…
We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for bra…
New link invariants derived from -Burau maps of braids.
problem Developing new link invariants from -Burau maps.
method Generalizing -Burau maps to all quotients of the group of the braid closure and proving corresponding -Alexander torsions.
result Obtained twisted -Alexander torsions of the braid closure, recovering topological information like hyperbolic volumes.
The paper constructs infinitely many prime hyperbolic knots.
problem Analyzing and constructing hyperbolic knots.
method Using plat position, braid group dynamics, and Heegaard splittings.
result Construction of infinitely many prime hyperbolic knots.
Researchers describe unitary representations of mixed braid groups.
problem Understanding unitary representations of mixed braid groups.
method Explicitly describe unitary representations on cohomology of Abelian branched covers.
result Image of the representation is generated by complex reflections and related to the multivariate Burau representation.
New Garside structures found for torus knot groups and related braid groups.
problem Finding Garside structures for torus knot groups and related braid groups.
method Introducing a new Garside monoid for -torus knot groups and other braid groups.
result New Garside structures for -torus knot groups and related braid groups are constructed.
On braids and groups math.GT
In [V.O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, arxiv:1501.05208] the first named author gave the definition of -free braid groups . Here we establish connections between free braid groups, classical braid groups and free groups: we describe e…
Paper constructs infinitely many non-braid positive hyperbolic L-space knots.
problem Proving the non-braid positivity of hyperbolic L-space knots.
method Constructs infinitely many hyperbolic L-space knots that are not braid positive.
result Distinct examples from Baker and Kegel's constructions.
We describe a series of complexes that relate to the braid groups as the matching complexes relate to the symmetric groups. A modified construction applies as well to other complexes based on edge sets in graphs. We show that our constructions will yield Cohen-Macauley complexes provided the underlying complexes are Co…
Following an idea of Gonçalvez, Guaschi and Ocampo on the usual braid group we construct crystallographic and Bieberbach groups as (sub)quotients of the generalized braid group associated to an arbitrary complex reflection group.
The study proves properties of specific groups acting on cube complexes.
problem Proving properties of specific groups acting on cube complexes.
method Elementary construction of a contractible cube complex.
result Proves are of type and is of type but not of type .
Invariants of braids found using shear coordinates in hyperbolic geometry.
problem Finding invariants of braids.
method Using shear coordinates in hyperbolic geometry.
result Developed a method for calculating braids invariants.
The relationships between braid ordering and the geometry of its closure is studied. We prove that if an essential closed surface in the complements of closed braid has relatively small genus with respect to the Dehornoy floor of the braid, is circular-foliated in a sense of Birman-Menasco's Braid foliation the…
We begin with a review of the notion of a braid group. We then discuss some known solutions to decision problems in braid groups. We then move on to proving new results in braid group algorithmics. We offer a quick solution to the generalized word problem in braid groups, in the special case of cyclic subgroups. We ill…
Boundary Braidsmath.GR
The n-strand braid group can be defined as the fundamental group of the configuration space of n unlabeled points in a closed disk based at a configuration where all n points lie in the boundary of the disk. Using this definition, the subset of braids that have a representative where a specified subset of these points …
We are concerned with orderable groups and particularly those with orderings invariant not only under multiplication, but also under a given automorphism or family of automorphisms. Several applications to topology are given: we prove that the fundamental groups of hyperbolic nonorientable surfaces, and the groups of c…
Positive braids with at least two twists form hyperbolic knots.
problem Classifying knots formed by specific braids.
method Conditions on positive braids with at least two full twists.
result Closure of such braids forms hyperbolic knots.
Study of wild mapping class groups and their cabled braids.
problem Understanding the structure of wild mapping class groups and their cabled versions.
method Define and study generalizations of pure -braid groups, establish product decompositions, and introduce fission trees.
result Obtain cabled versions of braid groups, related to braid operads.
Study automorphisms of pure braid groups on sphere homotopy groups.
problem Understanding automorphisms' effect on sphere homotopy groups.
method Examined Delta-group structure, proved invariance of cycle and boundary groups, computed action for few strands.
result Induced action of all automorphisms of pure braid groups on sphere homotopy groups.
We describe random walk boundaries (in particular, the Poisson--Furstenberg, or PF-boundary) for a vast family of groups in terms of the hyperbolic boundary of a special free subgroup. We prove that almost all trajectories of the random walk (with respect to an arbitrary nondegenerate measure on the group) converge to …