Title:

New invariants for groups

The principal part of this thesis starts with Chapter 2, Chapter 1 containing preliminary material. In Chapter 2, we give an exposition of the classical Alexander ideals of a group presentation whose set of generators is finite. These Alexander ideals are a group invariant; the chain of ideals calculated from presentations for isomorphic groups being equivalent. We also consider some classes of presentations whose groups cannot be distinguished by their Alexander ideals. In Chapter 3, we define a chain of ideals, the Bideals, which are calculated from a 3presentation with finite set of generators and relators. We show that these too are a group invariant and, moreover, that they can distinguish groups which the Alexander ideals cannot. In Chapter 4, we define for the class of groups of type FPn another new group invariant, the Enideals. These are calculated from a free resolution of type FPn for the group. We show that these generalise the Alexander and Bideals. The Enideals of a group are actually a special case of an invariant for group modules of type FPn. In the remainder of Chapter 4, we derive some properties of these module invariants and their equivalents for groups, including the connexion of these new invariants with the integral homology of a group. In Chapter 5, we consider the classes of modules and of groups whose Enideals are simple in a certain sense, the Etrivial modules and groups. In particular, we show that projective modules are Etrivial and, consequently, that groups of type FP are Etrivial. We consider how this relates to a question of Serre's concerning groups of type FP and of type FL. We then consider a larger class of groups, the E linked groups, whose Enideals are linked in adjacent dimensions in a certain sense. For a subclass of these groups we define an Euler characteristic, which extends the definition of Euler characteristics of Serre, Chiswell and Brown. We then study the closure properties of these classes of groups and the behaviour of the new Euler characteristic when graphs of these groups are constructed. Extensions of certain Etrivial groups are considered next, and we then demonstrate that, for every n > 1, the Eiideals can distinguish groups which have the same Eiideals for i < n and the same integral homology. In Chapter 6, we extend the definition of these new invariants to monoids and their modules, distinguishing a right and a lefthand version. We consider some of the properties of the monoid invariant, in particular, showing how the Enideals of certain groups can be obtained from those of a submonoid. Finally, the Enideals of monoids with a zero element are studied and we consider further the question of Serre.
