Asphericity of length 6 equations over torsion-free groups

It is a question of longstanding and current interest in combinatorial group theory to ask what happens to a given group when one builds a new group by adding some new generators and defining relations. That is, for a given group H and a set of words r(x), r ∈ r, involving a set of indeterminates x ∈ x the data P =< H, x : r > is called a relative presentation with coefficient group H; the group defined by this data is the quotient group For example, one can ask whether the natural homomorphism H → G is injective, an investigation known as equations over groups [9, 10, 12, 18]. A more general type of question asks how the structure of G is related to that of H. The concept of asphericity for relative presentations was introduced in [2] for the purpose of studying how the cohomology and finite subgroups of G are related to those of H.; In this dissertation we focus on relative presentations involving a single indeterminate x and a single relator r( x) of the form P=<H,x:xh1&ldots;xhn>, 1 where the elements h₁,…, h_n are taken from H. One of the early results, due to Levin [19], states that the natural homomorphism H → G is injective in this case. We show how to use the specific nature of Levin's argument to obtain new results in equations over groups.; Asphericity of the relative presentations (1) has been addressed by several authors [1, 2, 6, 14]. Scanning these results, one finds that for n ≤ 5, these presentations are always aspherical when H is torsion free. It is then a conse quence of the theory of aspherical relative presentations that if n ≤ 5 and H is torsion-free, then the new group G is also torsion-free as long as the relator is not a proper power in H ∗ F( x). The main result of this dissertation states that when n ≤ 6, the relative presentation (1) is aspherical whenever H is torsion-free and the relator is not a proper power.; The connection between asphericity and torsion-freeness goes deeper. The famous zero-divisor conjecture of Kaplansky [20] asserts that the integral group ring $Z$G of a torsion-free group G is a domain. A recent result of Ivanov [16] implies that a non-aspherical relative presentation of the form (1) that defines a torsion-free group G would provide a counterexample to Kaplansky's conjecture. Ivanov used a delicate argument involving spherical diagrams to prove his result. We provide a more conceptual treatment of the connection between zero-divisors and asphericity involving cellular homology and we show how a direct study of zero-divisors can yield results on asphericity of relative presentations.