Krull Dimension and Deviation in Certain Parafree Groups

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. G. 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 same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference μ(G)− n, where μ(G) is the minimum possible number of generators of G. Let G be fg; then Hom(G, SL(2,C)) inherits the structure of an algebraic variety, denoted by R(G), which is an invariant of fg presentations of G. 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 ≥ 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 ≥ 2, and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer. Structure of paper. New results in this paper are Theorem 1, Theorem 2, Theorem 3, Theorem 4, and Theorem 5. This paper is broken up into 3 parts: Introduction, Section one, Section two. In the introduction an outline of the mentality that guided this investigation is given, along with the proof of some preliminary results including the proof of Theorem 2, and Theorem 3. In Section One material involving sequences of primes, and groups associated with such sequences is developed. The section ends with a proof of Theorem 4. Section Two begins with a proof of Theorem 1, and ends with a proof of Theorem 5.