The group of parenthesized braids

We investigate a group $B\mathrm{\_}\bullet$ that includes Artin’s braid group $B\mathrm{\_}\mathrm{\infty }$ and Thompson’s group $F$. The elements of $B\mathrm{\_}\bullet$ 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 strech the distances between the strands. We prove that $B\mathrm{\_}\bullet$ 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 $B\mathrm{\_}\mathrm{\infty }$ into the automorphisms of a free group extends to $B\mathrm{\_}\bullet$.