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The group of parenthesized braids. (English) Zbl 1160.20027
Summary: We investigate a group $$B_\bullet$$ that includes Artin’s braid group $$B_\infty$$ and Thompson’s group $$F$$. The elements of $$B_\bullet$$ 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 $$B_\bullet$$ is a group of fractions, that it is orderable, admits a non-trivial self-distributive structure, i.e., one involving the law $$x(yz)=(xy)(xz)$$, embeds in the mapping class group of a sphere with a Cantor set of punctures, and that Artin’s representation of $$B_\infty$$ into the automorphisms of a free group extends to $$B_\bullet$$.

##### MSC:
 20F36 Braid groups; Artin groups 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms 20N02 Sets with a single binary operation (groupoids)
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