The Group of Parenthesized Braids

We investigate a group B• that includes Artin’s braid group B∞ 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 strech the distances between the strands. We prove that B• 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), it embeds in the mapping class group of a sphere with a Cantor set of punctures, and that Artin’s representation of B∞ into the automorphisms of a free group extends to B•. The aim of this paper is to study a certain group, denoted B•, which includes both Artin’s braid group B∞ [3, 9, 15] and Thompson’s group F [32, 28, 10]. The group B• is generated by (the copies of) B∞ and F , and its seemingly rich and deep properties appear to be a mixture of those of B∞ and F . Here, starting from a geometric approach in terms of parenthesized braid diagrams, we give an explicit presentation of B• that extends the standard presentations of B∞ and F , we prove that B• is a group of fractions, is an orderable group, and embeds into the mapping class group of a sphere with a Cantor set of punctures and into the automorphisms of a free group. Besides its group multiplication, B• is also equipped with a second binary operation satisfying the self-distributivity law x(yz) = (xy)(xz). We prove that every element of B• generates a free subsystem with respect to that second operation—which shows that the self-distributive structure of B• is highly non-trivial—and we deduce canonical decompositions for the elements of B•. The self-distributive structure is instrumental in proving most of the above results about the group structure of B•. Here the elements of B• are seen as parenthesized braids, which are braids in which the distances between the strands are not uniform. An ordinary braid diagram connects an initial sequence of equidistant positions to a similar final sequence, as for instance in where the initial and final set of positions can be denoted •••. A parenthesized braid diagram connects a parenthesized sequence of positions to another possibly different parenthezied sequence of positions, the intuition being that grouped positions are (infinitely) closer than ungrouped ones. An example is where the initial positions are (••)• and the final positions are •(••). Arranging such objects into a group leads to introducing, besides the usual braid generators σi that create crossings, new rescaling generators ai that shrink the distances between the strands in the vicinity of 1991 Mathematics Subject Classification. 20F36, 20N02, 57M25, 57S05.