arXiv research

New findings on algebraic structure of hyperbolic graph braid groups.

The paper classifies when certain graph braid groups are 3-manifold groups.

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 large-scale geometry of graph braid groups via cubical structures.

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 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 $L^2$-norm metric); this extends resul…

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk an…

We give formulae for the first homology of the $n$-braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the $n$-braid group over the graph is torsion-free and the conjectures about the first h…

New proof shows homomorphisms from pure braid groups to hyperbolic groups have cyclic images or factor through forgetful maps.

We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $\ge 5$. In order to have the necessity part, graphs are organized into small classes so that one of homological or cohomological characteristics of right-angled Artin groups can be applied. Fi…

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…

Paper constructs representations for virtual braids and flat braids.

Classifies certain graph 2-braid groups up to quasi-isometry.

The paper finds Artin presentations for the trivial group and identifies hyperbolic 3-braids.

We first show that the braid group over a graph topologically containing no $Θ$-shape subgraph has a presentation related only by commutators. Then using discrete Morse theory and triple Massey products, we prove that a graph topologically contains none of four prescribed graphs if and only if its 4-braid groups is a r…

We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$-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…

Study of Dehn filling quotients in hierarchically hyperbolic groups.

Study on planar graph braid groups' second homology.

The study finds a subgroup of graph braid groups that is a direct product of non-abelian free groups.

New insights into a complex hyperbolic braid group quotient.

New knot invariant from 3-braids and 6-valent graphs.

Graph braid groups' complexity stabilizes for most graphs.

Spatial graphs are decomposed into planar forests and braids.

We design an algorithm writing down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given graph. A key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and quadratic in the number of robots…

Geometrically describes hyperbolic structures on link complements using quantum groups.

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…

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 $Mod(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translat…

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.

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…

The paper defines parabolic subgroups for complex braid groups and proves they form a lattice.

In this article we calculate the n-string braid groups of certain non-contractible graphs. We use techniques from the work of A. Abrams, F. Connolly and M. Doig combined with Van Kampen's Theorem to prove these results.

In Garside groups, axes of Morse elements are strongly contracting.

The graph braid group of a complete bipartite graph is the fundamental group of a configuration space of points on the graph, which is a CAT(0) cube complex. We combine an analysis of the topology of links of vertices in this complex, the description of a hidden symmetry among the parameters, and known results from the…

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 …

Proves conjecture on graph configuration spaces' complexity.

The paper calculates asymptotic Betti numbers and homology multiplicities for graph configuration spaces.

Study quasi-isometry invariants of square complexes and their applications.

New groups $Γ_n^k$ connect braids and 3-manifolds.

This paper introduces cluster exchange groupoids for Coxeter-Dynkin diagrams and finds their fundamental groups are braid groups.

We prove that, in the $l$-ball of the Cayley graph of the braid group with $n \geqslant 3$ strands, the proportion of rigid pseudo-Anosov braids is bounded below independently of $l$ by a positive value.

We study a novel type of braid groups on a closed orientable surface $Σ$. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on $Σ$; a class of examples arises naturally in gauge theory, as moduli spaces of vortices in toric fibre bundles ove…

Study on embedding tree products into groups, distinguishing them.

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n $\ge$ 3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly as l tends to i…

The n-string braid group of a graph X is defined as the fundamental group of the n-point configuration space of the space X. This configuration space is a finite dimensional aspherical space. A. Abrams and R. Ghrist have conjectured that this braid group is a right angled Artin group if X is planar. We prove their conj…

New link invariants derived from $L^2$-Burau maps of braids.

The paper constructs infinitely many prime hyperbolic knots.

Configuration spaces of distinct labeled points on the plane are of practical relevance in designing safe control schemes for Automated Guided Vehicles (robots) in industrial settings. In this announcement, we consider the problem of the construction and classification of configuration spaces for graphs. Topological da…