Krull dimension and deviation in certain parafree groups

Abstract

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. Gilbert Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the lower central sequence with a free group of rank n. The deviation of a finitely generated (fg) parafree group G is the difference between the minimum possible number of generators of G and the rank of G. Let G be a fg group, then Hom(G,SL(2, C)) inherits the structure of an algebraic variety, denoted by R(G), and known as its "representation variety". If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G))=3n. It is known that for n \ge 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n \ge 2, and k \ge 1, there exist infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, it is shown that there exist infinitely many parafree groups G of rank n with Dim(R(G))> q, where q \ge 3n is an arbitrary integer.