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 B-ideals, which are calculated from a 3-presentation 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 En-ideals. These are calculated from a free resolution of type FPn for the group. We show that these generalise the Alexander and B-ideals. The En-ideals 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 En-ideals are simple in a certain sense, the E-trivial modules and groups. In particular, we show that projective modules are E-trivial and, consequently, that groups of type FP are E-trivial. 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 En-ideals 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 E-trivial groups are considered next, and we then demonstrate that, for every n > 1, the Ei-ideals can distinguish groups which have the same Ei-ideals 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 left-hand version. We consider some of the properties of the monoid invariant, in particular, showing how the En-ideals of certain groups can be obtained from those of a submonoid. Finally, the En-ideals of monoids with a zero element are studied and we consider further the question of Serre

Picture theory: algorithms and software

This thesis is concerned with developing and implementing algorithms based upon the geometry of pictures. Spherical pictures have been used in many areas of combinatorial group theory, and particularly, they have shown to be a useful method when studying the second homotopy module, 1T2, of a presentation ([3],[4],[7],[12],[41] and [64]). Computational programs that implement picture theoretical and design algorithms could advance the areas in which picture theory can be used, due to the much faster time taken to derive results than that of manual calculations. A variety of algorithms are presented. A data structure has been devised to represent spherical pictures. A method is given that verifies that a given data structure represents a picture, or set of pictures, over a group presentation. This method includes a new planarity testing algorithm, which can be performed on any graph. A computational algorithm has been implemented that determines if a given presentation defines a group extension. This work is based upon the algorithm of Baik et al. [1] which has been developed using the theory of pictures. A 3-presentation for a group G is given by , where P is a presentation for G and s is a set of generators for 1T2. The set s can be described in a number of ways. An algorithm is given that produces a generating set of spherical pictures for 1T2 when s is given in the form of identity sequences. Conversely, if s is given in terms of spherical pictures, then the corresponding identity sequences that describe 1T2 can be determined. The above algorithms are contained in the Spherical PIcture Editor (SPICE). SPICE is a software package that enables a user to manually draw pictures over group presentations and, for these pictures, call the algorithms described above. It also contains a library of generating pictures for the non abelian groups of order at most 30. Furthermore, a method has been implemented that automatically draws a spherical picture from a corresponding identity sequence. Again, this new graph drawing technique can be performed on any arbitrary graph