A Kirillov theory for divisible nilpotent groups.
The present paper is a continuation of our previous paper [Topology 44 (2005), 747-767], where we extended the Burau representation to oriented tangles. We now study further properties of this construction.
is the group presented by . In this paper, we study the structure of . We also give a new efficient presentation for the Projective Special Linear group and in particular we prove that is isomorphic to under certain conditions.
We prove that if is an integer and is a finitely generated soluble group such that every infinite set of elements of contains a pair which generates a nilpotent subgroup of class at most , then is an extension of a finite group by a torsion-free -Engel group. As a corollary, there exists an integer , depending only on and the derived length of , such that is finite. For , such depends only on .