The algebra of strand splitting. I. A braided version of Thompson’s group V

We construct a braided version BV of Thompson’s group V that surjects onto V . The group V is the third of three well known groups F , T and V created by Thompson in the 1960s that have been heavily studied since. See [6] and Section 4 of [5] for an introduction to Thompson’s groups. The group V is a subgroup of the homeomorphism group of the Cantor set C. It is generated by involutions [2, Section 12] and, if the metric on C is ignored, V can be viewed somewhat as a “Coxeter group” of permutations of C. In [4] we find presentations for BV and V that differ only in that the presentation for V has relations of the form x = 1 that are not present in the presentation for BV . Thus BV can be thought of as an “Artinification” of V . Our motivation for creating BV is a relationship between BV and the Thompson’s groups F and V on the one hand, and categories with multiplication on the other. Given a category C with multiplication, an isomorphism expressing associativity up to equivalence, and perhaps an isomorphism expressing commutativity up to equivalence, there are groups and epimorphisms