Pure braid subgroups of braided Thompson's groups

We describe some properties of braided generalizations of Thompson's groups, introduced by Brin and Dehornoy. We give slightly different characterizations of the braided Thompson's groups BV and (BV) over cap which lead to natural presentations which emphasize one of their subgroup-containment properties. We consider pure braided versions of Thompson's group F. These groups, BF and (BF) over cap, are subgroups of the braided versions of Thompson's group V. Unlike V, elements of F are order-preserving self-maps of the interval and we use pure braids together with elements of F thus again preserving order. We define these pure braided groups, give normal forms for elements, and construct infinite and finite presentations of these groups.