We prove that the braid group B4 on 4 strings, as well as its central quotient B4/〈z〉, have the property RD of Haagerup–Jolissaint. It follows that the automorphism group Aut(F2) of the free group F2 on 2 generators has property RD. We also prove that the braid group B4 is a group of intermediate rank (of dimension 3). Namely, we show that both B4 and its central quotient have exponential mesos...

We solve the isomorphism problem for braid groups on trees with n = 4 or 5 strands. We do so in three main steps, each of which is interesting in its own right. First, we establish some tools and terminology for dealing with computations using the cohomology of tree braid groups, couching our discussion in the language of differential forms. Second, we show that, given a tree braid group BnT on...

Let Γ be the fundamental group of the complement of a K(Γ, 1) hyperplane arrangement (such as Artin’s pure braid group) or more generally a homologically toroidal group as defined below. The subgroup of elements in the complex K-theory of BΓ which arises from complex unitary representations of Γ is shown to be trivial. In the case of real K-theory, this subgroup is an elementary abelian 2-group...