Authors: Carl Droms, Jacques Lewin and Herman Servatius
Reference: Journal of Pure and Applied Algebra 70, 251--261, 1991.
In
this paper we show that the unpermuted braid group on four strings is an
HNN-extension of the graph group F(S), where the graph S is the complement
of a path with 3 edges in the complete graph on 5 vertices.
The form of the extension resolves 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.
