arXiv research

Summary of pure cactus groups and circle points.

Explains the pure cactus group of degree three and its relation to four points on a circle.

Presented a simple group presentation for degree four cactus group.

Cactus doodles are geometric objects derived from cactus groups.

Affine cactus groups are CAT(0) and hyperbolic.

The BNSR-invariants of a group $G$ are a sequence $Σ^1(G)\supseteq Σ^2(G) \supseteq \cdots$ of geometric invariants that reveal important information about finiteness properties of certain subgroups of $G$. We consider the symmetric automorphism group $ΣAut_n$ and pure symmetric automorphism group $PΣAut_n$ of the free…

The main goal of this paper is a calculation of the integral (co)homology of the group of symmetric automorphisms of a free product. We proceed by giving a geometric interpretation of symmetric automorphisms via a moduli space of certain diagrams, which we name cactus products. To describe this moduli space a theory of…

Cactus improves auto-regressive decoding speed without sacrificing quality.

Graph theory criterion for Hodge theory to match linearly.

We find finite presentations for the automorphism group of the Artin pure braid group and the automorphism group of the pure braid group associated to the full monomial group.

Study automorphisms of pure braid groups on sphere homotopy groups.

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

In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups $P_n$, the welded pure braid groups $wP_n$, the virtual pure braid groups $vP_n$, as w…

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 $P_n$. We give different equivalents definitions for framed pure braid groups and we study exact sequences relating these groups with other generalisations of $P_n$, usually…

Corrects earlier work on surface orbifold pure braid groups.

In this paper it is proved that the pure braided Thompson's group BF admits a bi-order, analog to the bi-order of the pure braid groups.

Study of pure mapping class groups on infinite graphs.

Proves certain subgroups of genus 2 handlebody group are convex cocompact.

We show that the Artin pure braid group on at least four strands is not residually free. Our results also show that the pure braid group on at least three strands has corank two.

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.

Study of commutator subgroups and crystallographic quotients of virtual groups.

The question of whether a representation of Artin's pure braid group is faithful is translated to certain properties of the Lie algebra arising from the descending central series of the pure braid group, and thus the Vassiliev invariants of pure braids via work of T. Kohno \cite{kohno1,kohno2}. The main result is a Lie…

Consider the unit ball, $B = D \times [0,1]$, containing $n$ unknotted arcs $a_1, a_2, ..., a_n$ such that the boundary of each $a_i$ lies in $D \times \{0\}$. The Hilden (or Wicket) group is the mapping class group of $B$ fixing the arcs $a_1 \cup a_2 \cup ... \cup a_n$ setwise and fixing $D \times \{1\}$ pointwise. T…

It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology g…

Homomorphisms between pure mapping class groups are classified for certain genus surfaces.

Finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of ea…

Classifies surfaces for pure mapping class groups with automatic continuity.

The pure braid group cannot be realized as area-preserving homeomorphisms.

Paper studies pure virtual twin groups and their automorphisms.

Planar pure braids form a group that acts on a CAT(0) cubical complex.

Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.

In 1987 Bieri, Neumann and Strebel introduced a geometric invariant for discrete groups. In this article we compute and explicitly describe the BNS-invariant for the pure braid groups.

This article is dedicate to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group $VP_n$. Consequently we describe $VP_4$ as HNN-extension. As an application to classical braids, we find a new presentation of the Artin pure braid group $P_4$ in terms of the cabled gene…

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 subgroup behavior in genus-2 mapping class group identified.

We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is why we describe them as geometric presentaitons. Motivated by a presentation fo…

In this paper, we calculate the p-torsion of the Farrell cohomology for low genus pure mapping class groups with punctures, where p is an odd prime. Here, `low genus' means g=1,2,3; and `pure mapping class groups with punctures' means the mapping class groups with any number of punctures, where the punctures are not al…

We show that two knots have matching Vassiliev invariants of order less than n if and only if they are equivalent modulo the nth group of the lower central series of some pure braid group, thus characterizing Vassiliev's knot invariants in terms of the structure of the braid groups. We also prove some results about kno…

We show that any two elements of the pure braid group either commute or generate a free group, settling a question of Luis Paris. Our proof involves the theory of 3-manifolds and the theory of group actions on trees.

This work shows non-abelian quotient groups in string link concordance.

We adapt some of the methods of quantum Teichmüller theory to construct a family of representations of the pure braid group of the sphere.

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this…

Completes classification of G2-structures on specific nilpotent Lie groups.

A new invariant for pure braids is defined and shown not to be trivial.

We show that the problem of constructing a 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 ra…

This paper completes the classification of certain nilpotent Lie groups with specific geometric structures.

Classifies pure mapping class groups based on surface properties.

We show that the fundamental group of the space of ordered affine-equivalent configurations of at least five points in the real plane is isomorphic to the pure braid group modulo its centre. In the case of four points this fundamental group is free with eleven generators.