Algebraic Geometric Invariants of Parafree Groups

Abstract

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. 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 one-relator parafree groups are computed. It is then 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 in this paper it also shown that 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 > 1 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.