TREE GROUPS AND THE 4-STRING PURE BRAID GROUP

Given a graph GAMMA, undirected, with no loops or multiple edges, we define the graph group on GAMMA, F-GAMMA, as the group generated by the vertices of GAMMA, with one relation xy = yx for each pair x and y of adjacent vertices of GAMMA. In this paper we will show that the unpermuted braid group on four strings is an HNN-extension of the graph group F(S), where [GRAPHICS] The form of the extension will resolve a conjecture of Tits for the 4-string braid group. We will conclude, by analyzing the subgroup structure of graph groups in the case of trees, that for any tree T on a countable vertex set, F(T) is a subgroup of the 4-string braid group. We will also show that this uncountable collection of subgroups of the 4-string braid group is linear, that is, each subgroup embeds in GL(3, R), as well as embedding in Aut(F), where F is the free group of rank 2.