Invariants of Parafree Groups

Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL 2 C inherits the structure of an algebraic variety known as the representation variety of G in SL 2 C. This algebraic variety is an invariant of fg presentations of G. Call a group G parafree of rank n if it shares the lower central sequence with a free group of rank n, and if it is residually nilpotent. The deviation of a fg parafree group is the difference between the minimum possible number of generators of G and the rank of G. So parafree groups of deviation zero are actually just free groups. Parafree groups that are not free share a host of properties with free groups. In this paper algebraic geometric invariants involving the number of maximal irreducible components (mirc) of R(G), and the dimension of R(G) for certain classes of parafree groups are computed. It is shown that in an infinite number of cases these invariants successfully discriminate between isomorphism types within the class of parafree groups of the same rank. This is quite surprising, since an n generated group G is free of rank n iff Dim(R(G)) = 3n. In fact, a direct consequence of Theorem 1.6 in this paper is that given an arbitrary positive integer k, and any integer r ≥ 2 there exist infinitely many non-isomorphic fg parafree groups of rank r and deviation one with representation varieties of dimension 3r, having more than k mirc of dimension 3r. This paper also introduces many novel and surprising dimension formulas for the representation varieties of certain one-relator groups. General Structure of the Paper. This paper begins with an introduction where relevant ideas to what will follow are developed. It then goes on to define what a parafree group is, and how the notions that inspired G. Baumslag to give rise to such groups arose in the context of investigations conducted by W. Magnus, and a question of Hanna Neumann. The new results in this paper are Theorem 1.0, In Section One several results from the author's earlier work are introduced. In Section Two the following theorems are proven: 1.0, 1.1, and 1.6. The paper ends with a list notation.