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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)
20F36 Braid groups; Artin groups