On rigidity and the isomorphism problem for tree braid groups

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 $B_{n}T$ on $n=4$ or 5 strands, $H^{\ast }(B_{n}T)$ is an exterior face algebra. Finally, we prove that one may reconstruct the tree $T$ from a tree braid group $B_{n}T$ for $n=4$ or 5. Among other corollaries, this third step shows that, when $n=4$ or 5, tree braid groups $B_{n}T$ and trees $T$ (up to homeomorphism) are in bijective correspondence. That such a bijection exists is not true for higher dimensional spaces, and is an artifact of the 1-dimensionality of trees. We end by stating the results for right-angled Artin groups corresponding to the main theorems, some of which do not yet appear in the literature.