On generalized braid groups

We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show that the parametrized braid group is isomorphic to the semi-direct product of the Steinberg group (of the ring) with the classical braid group. The technical heart of the proof is the Pure Braid Lemma, which asserts that certain elements of the parametrized braid group commute with the pure braid group. This first part treats the case of the root system A_n ; in the second part we prove a similar theorem for the root system D_n .

The singular braids with [Formula: see text] strands, [Formula: see text], were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by [Formula: see text]. There has been another generalization of braid groups, denoted by [Formula: see text], [Formula: see text], which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group [Formula: see text] simultaneously generalizes the classical braid group, as well as the virtual braid group on [Formula: see text] strands. We investigate the commutator subgroups [Formula: see text] and [Formula: see text] of these generalized braid groups. We prove that [Formula: see text] is finitely generated if and only if [Formula: see text], and [Formula: see text] is finitely generated if and only if [Formula: see text]. Further, we show that both [Formula: see text] and [Formula: see text] are perfect if and only if [Formula: see text].

We recall that the two-sided crossed product of finite cyclic groups is actually a generalization of the crossed product construction of the same type of groups (cf. [10]). In this paper, by considering the crossed and two-sided crossed products obtained from both finite and infinite cyclic groups, we first present the complete rewriting systems and normal forms of elements over crossed products. (We should note that the complete rewriting systems and normal forms of elements over two-sided crossed products have been recently defined in [10]). In the crossed product case, we will consider their presentations that were given in [2]. As a next step, by using the normal forms of elements of these two products, we calculate the growth series of the crossed product of different combinations of finite and infinite cyclic groups as well as the growth series of two-sided crossed product of finite cyclic groups.

Schur rings over the infinite dihedral group Z ⋊ Z 2 are studied according to properties of Schur rings over infinite groups and the classification of Schur rings over infinite cyclic groups. Schur rings over Z ⋊ Z 2 are classified under the assumption that Z is an A -subgroup. Those Schur rings are proved to be traditional.

We study the problem of finding generators for the fundamental group G of a\nspace of the following sort: one removes a family of complex hyperplanes from n\ndimensional complex vector space, or n dimensional complex hyperbolic space, or\nthe Hermitian symmetric space for O(2,n), and then takes the quotient by a\ndiscrete group $P{\\Gamma}$. The classical example is the braid group, but there\nare many similar "braid-like" groups that arise in topology and algebraic\ngeometry. Our main result is that if $P{\\Gamma}$ contains reflections in the\nhyperplanes nearest the basepoint, and these reflections satisfy a certain\nproperty, then G is generated by the analogues of the generators of the\nclassical braid group. We apply this to obtain generators for G in a particular\nintricate example in complex hyperbolic space of dimension 13. The interest in\nthis example comes from a conjectured relationship between this braid-like\ngroup and the monster simple group M, that gives geometric meaning to the\ngenerators and relations in the Conway-Simons presentation of $(M \\times M):2$.\n

Configuration Lie groupoids and orbifold braid groups

Infinite cyclic groups created by various objects belong to the class to the class basic algebraic structures. In this paper, we construct the infinite cyclic group of differential neurons which are modifications of artificial neurons in analogy to linear ordinary differential operators of the n-th order. We also describe some of their basic properties.

Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link

A new construction of a blind signature scheme based on braid groups is proposed. In the random oracle model, the proposed scheme is provably unforgeable against chosen message attacks, assuming that the one-more matching conjugate problem in braid groups is intractable. Furthermore, in the infinite group model, the scheme is proved to be perfectly blind. Our construction represents a technique to lift a braid group to its conjugate subgroups for particular applications. The proposed scheme is very fast in signing but relatively slow in verifying and is thus suitable for scenarios that require signing as soon as possible but permit a slight delay in verifying. In addition, our proposal is invulnerable to known quantum attacks and therefore would be a good alternative to RSA-based and DLP-based blind signatures in the post-quantum era.

From representations of the braid group to solutions of the Yang-Baxter equation

The study of homology groups ${\mathfrak {B}_n}(\Pi ,A),\mathfrak {B}$ a variety, $\Pi$ a group in $\mathfrak {B}$, and $A$ a suitable $\Pi$-module, is continued. A ’Tor’ is constructed which gives a better (but imperfect) approximation to these groups than a Tor previously considered. ${\mathfrak {B}_2}(\Pi ,Z)$ is calculated for various varieties $\mathfrak {B}$ and groups $\Pi$.

For each integer n ≥ 2 , simple noetherian hereditary rings R are constructed such that R has uniform (Goldie) dimension n but R is: not isomorphic to a matrix ring over an integral domain. That R is not a matrix ring is verified K-theoretically, using Quillen’s machinery [3] to compute K0(R). In fact, for these examples K0(R) is free abelian of rank n.

On the computational complexity of the theory of Abelian groups

Let (W, S) be an infinite Coxeter system. To each geometric representation of W is associated a root system. While a root system lives in the positive side of the isotropic cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the set E of limit points of the directions of roots. In this article we study the close relations of the imaginary cone with the set E, which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular, we show that the W-action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of W, i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.

It is proved that Thom spectra of generalized braid groups are the wedges of suspensions over the Eilenberg–MacLane spectrum for ℤ/2. The precise structure of the Thom spectra of the generalized braid groups of the types C and D is obtained. For the generalized braid groups of type C the natural pairing analogous to the pairing of the classical braids is studied. This pairing generates the multiplicative structure of the Thom spectrum such that the corresponding bordism theory has the coefficient ring isomorphic to the polynomial ring over ℤ/2 on one generator of dimension 1[ratio ]ℤ/2[ s ].