Groups with One Defining Relation

Cuny City College, New York NY

Investigators

Abstract

The principal investigator proposes here to study groups defined by a single relation. Such groups arise as fundamental groups in algebraic topology, as discrete groups of transformations in analysis and geometry and in abstract group theory. The importance of such a study is in its many connections to other branches of mathematics, as well as to computer science and logic. Progress will shed light on a number of other subjects. These include the theories of hyperbolic and automatic groups, computability, genetic and other algorithms and may lead to a deeper understanding of finitely presented groups as a whole. This proposal represents a first step in an overall study of one-relator groups, using the number of steps in a so-called Magnus breakdown, as the basis for induction. The principal investigator, together with his many colleagues, will initiate this study by focussing attention on one-relator groups of planarity one. These are the very special HNN extensions of a free group that arise in the first step of the Magnus breakdown referred to above. Groups are mathematical structures which are designed, in part, to capture the intrinsic nature of symmetry. Consequently groups play an extremely important role in geometry, in chrystallography, in particle physics, in chemistry and in cryptography as well as in much of mathematics. Their uses in cryptography are yet to be exploited; RSA encryption may be viewed as a first step in this direction. Many of the problems of the physical world can be translated into problems in group theory. On the other hand, many processes, such as evolution, can be used to unravel the nature of particular groups. Thus group theory provides a testing ground for a better understanding of these processes. The principal investigator, in the work to be undertaken, will, in particular, use group theory to attempt to improve on one of the most exciting new aspects of computer science, so-called genetic algorithms. Such algorithms appear to be a means for designing ever more intelligent computers.