The group of parenthesized braids

By Patrick Dehornoy. 2004

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We investigate a group B? that includes Artin's braid group B8 and Thompson's group F. The elements of B? are represented by braids diagrams in which the distances between the strands are not uniform and, besides the usual crossing generators, new rescaling operators shrink or stretch the distances between the strands. We prove that B? is a group of fractions, that it is orderable, admits a nontrivial self-distributive structure, i.e., one involving the law x(yz)=(xy)(xz), it embeds in the mapping class group of a sphere with a Cantor set of punctures, and that Artin's representation of B8 into the automorphisms of a free group extends to B?.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4db111e0b9bdb78d2777d90330202a6674102399

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Geometric presentations for Thompson's groups (2004, background): “tions; braid groups. 1 2 PATRICK DEHORNOY work can be seen as an independent rediscovery of these groups. Let us mention still another approach to B• as a group of so-called parenthesized braids: see [9], which contains a thorough study of B•. Various groups connecting Thompson’s groups and braids, ”
The braided Ptolemy-Thompson group is finitely presented (2005, background): “raids B∞ on inﬁnitely many strands. Notice that an extension BV of the larger Thompson group V by a pure braid group on inﬁnitely many strands has been recently considered by M. Brin and P. Dehornoy ([4, 5, 12, 13]). It constitutes the planar counterpart of the group B, in which the Cantor surface i”