Groups with One Defining Relation

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The principal investigator proposes here to study groups defined by a single relation.Such groups arise as fundamental groups in algebraic topology, as discretegroups of transformations in analysis and geometry and in abstract grouptheory. The importance of such a study is in its many connections to otherbranches of mathematics, as well as to computer science and logic. Progress will shedlight 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 awhole. This proposal represents a first step in an overall study of one-relatorgroups, using the number of steps in a so-called Magnus breakdown, as thebasis 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 theMagnus breakdown referred to above. Groups are mathematical structures which are designed, in part, to capture theintrinsic nature of symmetry. Consequently groups play an extremely important rolein geometry, in chrystallography, in particle physics, in chemistry and in cryptographyas 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. Thusgroup 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 computerscience, so-called genetic algorithms. Such algorithms appear to be a means for designing ever more intelligent computers.