Combinatorial properties of virtual braids

We determine the lower central series of the virtual braid group VBn and of the kernels of two different projections of VBn in Sn: the normal closure of the Artin braid group Bn, that we will denote by Hn, and the so-called virtual pure braid group VPn, which is related to Yang Baxter equation. We describe relations between Hn and VPn and we provide a connection between virtual pure braids and the finite type invariant theory for virtual knots defined by Goussarov, Polyak and Viro.     

On the inverse braid monoid

Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial n may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely we prove an inclusion into a monoid of partial monomorphisms of a free group. This gives a solution of the word problem. Another solution is obtained by an approach similar to that of Garside. We give also the analogues of Artin presentation with two generators and Sergiescu graph-presentations.     

Embedding right-angled Artin groups into graph braid 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 group.      

Boundary amenability of relatively hyperbolic groups

Let K be a fine hyperbolic graph and Γ be a group acting on K with finite quotient. We prove that Γ is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group Γ acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations.     

The <Emphasis Type="Italic">K</Emphasis>(π,1)-conjecture for the affine braid groups

The complement of the hyperplane arrangement associated to the (complexified) action of a finite, real reflection group on ⁿ is known to be a K(,1) space for the corresponding Artin group $\Cal A$. A long-standing conjecture states that an analogous statement should hold for infinite reflection groups. In this paper we consider the case of a Euclidean reflection group of type à _n and its associated Artin group, the affine braid group $\tilde{\Cal A}$. Using the fact that $\tilde{\Cal A}$ can be embedded as a subgroup of a finite type Artin group, we prove a number of conjectures about this group. In particular, we construct a finite, $n$-dimensional K(,1)-space for $\tilde{\Cal A}$, and use it to prove the K(,1) conjecture for the associated hyperlane complement. In addition, we show that the affine braid groups are biautomatic and give an explicit biautomatic structure.     

Presentations of subgroups of the braid group generated by powers of band generators

According to the Tits conjecture proved by Crisp and Paris (2001) [4], the subgroups of the braid group generated by proper powers of the Artin elements σi are presented by the commutators of generators which are powers of commuting elements. Hence they are naturally presented as right-angled Artin groups.The case of subgroups generated by powers of the band generators aij is more involved. We show that the groups are right-angled Artin groups again, if all generators are proper powers with exponent at least 3. We also give a presentation in cases at the other extreme, when all generators occur with exponent 1 or 2. Such presentations are distinctively more complicated than those of right-angled Artin groups.     

The E-theoretic descent functor for groupoids

The paper establishes, for a wide class of locally compact groupoids Γ, the E-theoretic descent functor at the C^∗-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. Section 2 shows that Γ-actions on a C₀(X)-algebra B, where X is the unit space of Γ, can be usefully formulated in terms of an action on the associated bundle B^?. Section 3 shows that the functor B→C^∗(Γ,B) is continuous and exact, and uses the disintegration theory of J. Renault. Section 4 establishes the existence of the descent functor under a very mild condition on Γ, the main technical difficulty involved being that of finding a Γ-algebra that plays the role of Cbcont(T,B) in the group case.     

Toric complexes and Artin kernels

A simplicial complex L on n vertices determines a subcomplex TL of the n-torus, with fundamental group the right-angled Artin group GL. Given an epimorphism χ:GL→Z, let be the corresponding cover, with fundamental group the Artin kernel Nχ. We compute the cohomology jumping loci of the toric complex TL, as well as the homology groups of with coefficients in a field k, viewed as modules over the group algebra kZ. We give combinatorial conditions for to have trivial Z-action, allowing us to compute the truncated cohomology ring, . We also determine several Lie algebras associated to Artin kernels, under certain triviality assumptions on the monodromy Z-action, and establish the 1-formality of these (not necessarily finitely presentable) groups.     

Recognition of Virtual Braids

The virtual braid group VB_n containing the Artin braid group B_n as a subgroup is studied. A familiar complete invariant on B_n is extended to VB_n. It is conjectured that the new invariant is also complete. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 267–286.     

Periodic elements in Garside groups

Let G be a Garside group with Garside element Δ, and let Δ^m be the minimal positive central power of Δ. An element g∈G is said to be periodic if some power of it is a power of Δ. In this paper, we study periodic elements in Garside groups and their conjugacy classes.We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of G is cyclic; if g^k=Δ^ka for some nonzero integer k, then g is conjugate to Δ^a; every finite subgroup of the quotient group G/〈Δ^m〉 is cyclic.By a classical theorem of Brouwer, Kerékjártó and Eilenberg, an n-braid is periodic if and only if it is conjugate to a power of one of two specific roots of Δ². We generalize this to Garside groups by showing that every periodic element is conjugate to a power of a root of Δ^m.We introduce the notions of slimness and precentrality for periodic elements, and show that the super summit set of a slim, precentral periodic element is closed under any partial cycling. For the conjugacy problem, we may assume the slimness without loss of generality. For the Artin groups of type , , , and the braid group of the complex reflection group of type (e,e,n), endowed with the dual Garside structure, we may further assume the precentrality.     

Vines and partial transformations

We introduce the partial vine monoid PVn. This monoid is related to the partial transformation semigroup PTn in the same way as the braid group Bn is related to the symmetric group Sn, and contains both the vine monoid [T.G. Lavers, The theory of vines, Comm. Algebra 25 (4) (1997) 1257-1284] and the inverse braid monoid [D. Easdown, T.G. Lavers, The inverse braid monoid, Adv. Math. 186 (2) (2004) 438-455]. We give a presentation for PVn in terms of generators and relations, as well as a faithful representation in a monoid of endomorphisms of a free group. We also derive a new presentation for PTn.     

Reflection groups of geodesic spaces and Coxeter groups

H.S.M. Coxeter showed that a group Γ is a finite reflection group of an Euclidean space if and only if Γ is a finite Coxeter group. In this paper, we define reflections of geodesic spaces in general, and we prove that Γ is a cocompact discrete reflection group of some geodesic space if and only if Γ is a Coxeter group.     

Subnormal structure of non-stable unitary groups over rings

Let R be a commutative ring with identity in which 2 is invertible. Let H denote a subgroup of the unitary group U(2n,R,Λ) with n≥4. H is normalized by EU(2n,J,Γ^J) for some form ideal (J,Γ^J) of the form ring (R,Λ). The purpose of the paper is to prove that H satisfies a “sandwich” property, i.e. there exists a form ideal (I,Γ^I) such that

EU(2n,IJ⁸Γ^J,Γ)⊆H⊆CU(2n,I,Γ^I).     

Algebraic invariants for right-angled Artin groups

A finite simplicial graph Γ determines a right-angled Artin group G_Γ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of G_Γ, directly from the graph Γ. Partially supported by NSF grant DMS-0311142.     

Braids and partial permutations

We study the inverse braid monoid IBn introduced by Easdown and Lavers in 2004. We completely describe the factorizable structure of IBn and use this to give a new proof of the Easdown-Lavers presentation; we also derive several new presentations, each of which gives rise to a new presentation of the symmetric inverse monoid. We then define and study the pure inverse braid monoid IPn which is related to IBn in the same way that the pure braid group is related to the braid group.     

On loxodromic actions of Artin–Tits groups

Artin–Tits groups act on a certain delta-hyperbolic complex, called the “additional length complex”. For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mapping class groups, we conjecture that “most” elements of Artin–Tits groups act loxodromically. More precisely, in the Cayley graph of a subgroup G of an Artin–Tits group, the proportion of loxodromically acting elements in a ball of large radius should tend to one as the radius tends to infinity. In this paper, we give a condition guaranteeing that this proportion stays away from zero. This condition is satisfied e.g. for Artin–Tits groups of spherical type, their pure subgroups and some of their commutator subgroups.     

Representing abstract groups by powers of a graph

We deal with the problem of representing several abstract groups simultaneously by one graph as automorphism groups of its powers. We call subgroups Γ₁,…, Γ_n of a finite group Γ representable iff there is a graph G and an injective mapping φ from ∪_i=1ⁿΓ_i into the symmetric group on V(G) such that for i=1,…, n φ|_Γi is a monomorphism onto Aut Gⁱ. We give a necessary and a sufficient condition for groups being representable, the latter implying, e.g., that finite groups Γ₁≤…≤Γ_n are representable.     

A special class of tensor categories initiated by inverse braid monoids

In this paper, we introduce the concept of a wide tensor category which is a special class of a tensor category initiated by the inverse braid monoids recently investigated by Easdown and Lavers [The Inverse Braid Monoid, Adv. in Math. 186 (2004) 438-455].The inverse braid monoidsIB_n is an inverse monoid which behaves as the symmetric inverse semigroup so that the braid group B_n can be regarded as the braids acting in the symmetric group. In this paper, the structure of inverse braid monoids is explained by using the language of categories. A partial algebra category, which is a subcategory of the representative category of a bialgebra, is given as an example of wide tensor categories. In addition, some elementary properties of wide tensor categories are given. The main result is to show that for every strongly wide tensor category C, a strict wide tensor category C^str can be constructed and is wide tensor equivalent to C with a wide tensor equivalence F.As a generalization of the universality property of the braid category B, we also illustrate a wide tensor category through the discussion on the universality of the inverse braid category IB (see Theorem 3.3, 3.6 and Proposition 3.7).     

K(π, 1) and word problems for infinite type Artin–Tits groups, and applications to virtual braid groups

Let Γ be a Coxeter graph, let (W, S) be its associated Coxeter system, and let (A, Σ) be its associated Artin–Tits system. We regard W as a reflection group acting on a real vector space V. Let I be the Tits cone, and let E _Γ be the complement in I + iV of the reflecting hyperplanes. Recall that Salvetti, Charney and Davis have constructed a simplicial complex Ω(Γ) having the same homotopy type as E _Γ. We observe that, if ${T \subset S}$ , then Ω(Γ _T ) naturally embeds into Ω (Γ). We prove that this embedding admits a retraction ${\pi_T: \Omega(\Gamma) \to \Omega (\Gamma_T)}$ , and we deduce several topological and combinatorial results on parabolic subgroups of A. From a family ${\mathcal{S}}$ of subsets of S having certain properties, we construct a cube complex Φ, we show that Φ has the same homotopy type as the universal cover of E _Γ, and we prove that Φ is CAT(0) if and only if ${\mathcal{S}}$ is a flag complex. We say that ${X \subset S}$ is free of infinity if Γ _X has no edge labeled by ∞. We show that, if ${E_{\Gamma_X}}$ is aspherical and A _X has a solution to the word problem for all ${X \subset S}$ free of infinity, then E _Γ is aspherical and A has a solution to the word problem. We apply these results to the virtual braid group VB _n . In particular, we give a solution to the word problem in VB _n , and we prove that the virtual cohomological dimension of VB _n is n?1.     

Toeplitz operators associated to commuting row contractions

Let H be a complex Hilbert space and let {Tn}_n?1 be a sequence of commuting bounded operators on H such that . Let denote the space of all operators X in B(H) for which and suppose that . We will show that there exists a triple {K,Γ,{Un}_n?1} where K is a Hilbert space, Γ:K→H is a bounded operator and {Un}_n?1⊂B(K) is a sequence of commuting normal operators with such that TnΓ=ΓUn for n?1, and for which the mapping Y?ΓYΓ^∗ is a complete isometry from the commutant of {Un}_n?1 onto the space . Moreover we show that the inverse of this mapping can be extended to a ^∗-homomorphism