Graph products of groups

Abstract

In the 1970's Baudisch introduced the idea of the semifree group, that is, a group in which the only relators are commutators of generators. Baudisch was mainly concerned with subgroup problems, employing length arguments on the elements of these groups. More recently Droms and Servatius have continued the study of semifree, or graph groups, as they call them. They answer some of the questions left open by the work of Baudisch. It is possible to take the graph analogy a level higher and study graph products of groups, which not only generalise graph groups, but also free and direct products. In this thesis we seek to explore the properties of graph products of groups. After some preliminaries in chapter 1, chapter 2 quotes the main results from the work of Baudisch, Droms and Servatius on graph groups, and includes a few elementary results. In chapter 3 we show that many of the well known results about free products and direct products will generalise to graph products. We also extend some of the results on graph groups and give a counter example to a plausible conjecture. We develop a normal form for elements in graph products and, with the use of a generalised free product representation, show solvability of the word and conjugacy problems. In-chapter 4 we examine the concept of graphological indecomposability. Having disposed of an obvious conjecture by way of a counter example we present a number of isomorphism theorems. Chapter 5 is devoted to residual properties of graph products. Much work has been done by Stebe, Allenby and others on residual finiteness, conjugacy separability and potency of free groups and free products. We generalise some of these results. Finally, in chapter 6 we return to graph groups for a look at the Freiheitssatz. Various subclasses have been covered by Pride, Baumslag and Howie, and we seek to extend their results