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The intersection of flat subsets of a braid group. (English) Zbl 0962.20027
Cossey, John (ed.) et al., Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, July 14-19, 1996. Berlin: de Gruyter. 139-146 (1999).
With an $$(n-1)$$-tuple $$(k_1,\dots,k_{n-1})$$ of integers the author associates a braid which is called flat braid. The full subgraph of the Cayley graph of $$B_n$$ generated by the flat braids can be regarded as a graph embedded in $$\mathbb{R}^{n-1}$$, whose set of vertices is $$\mathbb{Z}^{n-1}$$, and whose edges have length 1 and are parallel to the axes. For $$i=2,\dots,n$$, let $$\Delta_i$$ denote the positive braid which generates the center of the braid group $$B_i$$, viewed as a subgroup of $$B_n$$. Then the set of pure flat braids is the free Abelian group of rank $$n-1$$ generated by $$\{\Delta_2,\dots,\Delta_n\}$$ and denoted by $$A$$. Define a diagonal subgroup to be a subgroup of $$A$$ generated by a subset of $$\{\Delta_2,\dots,\Delta_n\}$$, and let $$F$$ denote the set of flat braids. The main result of this paper is that, for all $$x,y\in B_n$$, there exist $$z\in B_n$$, a finite subset $$X\subset B_n$$, and a diagonal subgroup $$D\subset A$$, such that $$xF\cap yF=zDX$$.
For the entire collection see [Zbl 0910.00040].
Reviewer: Luis Paris (Dijon)
##### MSC:
 20F36 Braid groups; Artin groups
##### Keywords:
Cayley graphs; flat braids; centers; braid groups