Davidson, Peter John
Geometric methods in the study of Pride groups and relative presentations.
PhD thesis, University of Glasgow.
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Combinatorial group theory is the study of groups given by
presentations. Algebraic and geometric methods pervade this area of mathematics and it is the latter which forms the main theme of this thesis. In particular, we use diagrams and pictures over presentations to study problems in the domain of finitely presented groups. Our thesis is split into two distinct halves, though the techniques used in each are very similar. In Chapters 2 - 4 we study Pride groups with the aim to solve their word and conjugacy problems. We also study the second homotopy module of a natural presentation of a Pride group. Chapters 6 and 7 are devoted to the study of relative presentations, with particular attention being paid to those of the form < H, t ; t^n a t^-1 b>. Determining when such presentations are aspherical is our main objective.
Chapter 1 covers the basic material that is used throughout this thesis. The main topics of interest are free groups; presentations of groups; the word, conjugacy, and isomorphism problems for finitely presented groups; first and second order Dehn functions of finitely presented groups; diagrams and pictures over finite presentations; and the second homotopy module of a finite presentation. The reader may skip Chapter 1 if they are familiar with this material.
A Pride group is a finitely presented group which can be defined by means of a finite simplicial graph; this is done in Chapter 2. Examples of Pride groups are given in Section 2.1. This section also contains the statements of Conditions (I), (II), (H-I), (H-II), and the asphericity condition. We will always assume that a Pride group satisfies one of these conditions. In Section 2.2 we survey the known results that appear in the literature, while in Section 2.3 we present
our original results. We obtain isoperimetric functions for a vertex-finite Pride group G which satisfies (I), (II), (H-I) or (H-II). Sufficient conditions are then obtained for G to have a soluble word problem. Solutions of the conjugacy problem for G are also obtained. However, we require that G satisfies some extra conditions. We calculate a generating set for the second homotopy module of the natural presentation of a non-spherical Pride group, i.e. one which satisfies the asphericity condition. Using this generating set, we obtain an upper bound for the second order Dehn function of a non-spherical vertex-free Pride group. We
also obtain information about the second order Dehn function of an arbitrary non-spherical Pride group.
Chapter 3 contains various technical results that are needed in Chapter 4. The main focus is that of diagrams over the standard presentation of a vertex-finite Pride group. We study simply-connected r-diagrams in Section 3.1 and in Section 3.2 we study annular r-diagrams. Propositions 3.1.1, 3.2.1, 3.2.2, and Theorems 3.2.1 and 3.2.2 are the main results of this chapter.
Chapters 4 and 5 are devoted to the proofs of our main results. Proofs of our results for the word and conjugacy
problems of a vertex-finite Pride group are contained in Chapter 4, while Chapter 5 contains proofs of our
results about the second homotopy module of a non-spherical Pride group. Chapter 5 also contains a study of pictures over the natural presentation of such a group.
In Chapter 6, we turn our attention to relative presentations. Our interest lies in determining when such presentations are aspherical. Relevant background material and definitions are given in this chapter and pictures over relative presentations are also studied. Five tests which are used to determine whether or not a relative presentation
is aspherical are given in Section 6.4. Chapter 6 also contains a brief survey of known results in this area.
In Chapter 7, the final chapter of this thesis, we present our original contribution to the area of aspherical relative presentations. In particular, we determine when the relative presentation < H, t ; t^n a t^-1 b > is aspherical where n is greater than or equal to 4 and a, b are elements of H each of order at least 3. There are four exceptional cases for which asphericity cannot be determined.
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