Tree groups and the 4-string pure braid group

Given a graph Г, undirected, with no loops or multiple edges, we define the graph group on Г, F_Г, as the group generated by the vertices of Г, with one relation xy = xy for each pair x and y of adjacent vertices of Г.

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

S =

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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, ), as well as embedding in Aut(F), where F is the free group of rank 2.