Artin’s braids, braids for three space, and groups Γn4 and Gnk

Abstract

We construct a group Γ 4 n corresponding to the motion of points in R 3 from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on n strands to the product of copies of Γ 4n . We will also study the group of pure braids in R 3 , which is described by a fundamental group of the restricted configuration space of R 3 , and define the group homomorphism from the group of pure braids in R 3 to Γ 4 n . In the end of this paper we give some comments about relations between the restricted configuration space of R 3 and triangulations of the 3-dimensional ball and Pachner moves.1991 Mathematics Subject Classification. 57M25, 57M27.

Manifolds of Triangulations, braid groups of manifolds and the groups $Γ_{n}^{k}$

“…Thus, we characterise the local structure of the manifold of triangulations and define groups Γ k n to describe it. These groups extend the construction of the group Γ 4 n considered in [5]. The paper concludes with the main results in the third section including Γ k n -valued invariant of braids on manifolds, an invariant in a direct product of the groups Γ k n , a description of triangulations of a polyhedron in terms of Γ k n , and a description of a variation on the Γ k n groups in geometrical sense corresponding to oriented triangulations.…”

“…In the present paper we deal mostly with the case of d ≥ 3. The very interesting initial case d = 2 is studied in [5].…”

“…Our definition of Γ 4 n differs slightly from the definition of these groups in [5]. To obtain the groups defined in [5] we should add relations a 2 m1m2,m3m4 = 1 and a m1m2,m3m4 a m1m2,m5m6 = a m1m2,m5m6 a m1m2,m3m4 , a m1m2,m3m4 a m1m5,m2m6 = a m1m5,m2m6 a m1m2,m3m4 ,…”

Word and Conjugacy Problems in Groups $G_{k+1}^{k}$

“…For G k n groups are closely related to braid groups, it is natural to pose the word problem and the conjugacy problem for them. The connection of the groups G k n with fundamental groups of manifolds, with Coxeter groups and other geometrical and algebraic structures was widely studied (see, for example, [8,11,13,14], see also [1], where braids with points were introduced, which are closely related to the G 2 n group). The class of groups for n = k + 1 is especially interesting because of the absence of the far commutativity relation.…”