The relation gap problem

Introduction The augmentation ideal IG of a group can be thought of as a linear model of G. A presentation of G gives rise to a ZG -module presentation of IG. In particular, if G has an n -generator m -relator presentation, then so does IG. Natural questions ar 1 Can IG be generated by fewer elements than needed to generate G? 2 If a presentation of G requires m relators, can one have fewer than m relators in the corresponding presentation of IG? Karl Gruenberg called the difference between the minimal numbers of generators of G and the minimal number of generators of the left ZG -module IG the generation gap of G, and referred to the first question as the generation gap problem. He called the difference between the minimal number of normal generators of N and the the minimal number of generators of the left module N /[N, N] the relation gap of F /N, and referred to the second question as the relation gap problem. The generation gap problem was solved in 1974 by Cossey, Gruenberg, and Kovacs. They showed that arbitrarily large generation gaps can occur for finite groups. The relation gap problem for finitely presented groups, however, is still open. The mathematics employed in search of an answer ranges from representation theory and homological group theory to techniques in geometric group theory and topology. Two results deserve special mention, since they present, in a sense, extreme cases of the problem. The first says that for finite groups the rank of the relation module can be computed and the difference is a group invariant and does not depend on the choice of the finite presentation F /N. Thus, for finite groups, the relation gap question comes down to the question of whether the difference is a group invariant. The second result says that infinite relation gaps can occur for finitely generated infinite groups. This was shown by Bestvina and Brady in the early 1990's by developing geometric techniques such as Morse theory in the context of cubical complexes.