On rigidity and the isomorphism problem for four strand tree braid groups
by
Lucas Sabalka
University of Illinois at Urbana-Champaign

Given a tree braid group B_nT on n = 4 strands, we are able to reconstruct the tree T. Thus tree braid groups B₄T 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 use this bijection to solve a version of the isomorphism problem for tree braid groups with n = 4 strands. We also make some comments on the possibility of generalizing this solution to tree braid groups with more strands.

Date received: February 3, 2006