Mathematical Sciences: Discrete Groups and Quasiconformal Mappings

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ABSTRACT Proposal: DMS-9622808 PI: Gehring This project addresses problems concerning discrete groups of Mobius transformations, 3-manifolds and orbifolds, quasiconformal mappings, and function theoretic properties of plane domains. For example, the Gehring plans to continue a joint study with G. J. Martin of the set of values which cannot be assumed by the trace of the commutator of pairs of elements of a nonelementary discrete group. This study has already yielded a great deal of information about the minimum possible volumes of 3-manifolds and orbifolds and it should, in particular, allow one to prove that the orbifold associated with the 3-5-3 hyperbolic tetrahedral group has minimum volume among all 3-manifolds and orbifolds. The main tools for this study are: (1) complex iteration to identify big filled in Julia sets E for a special family P of polynomials, and (2) a covering argument using the family P and the filled in Julia sets E to obtain large regions D of excluded values for the traces of the commutators of pairs of elements. The Gehring hopes also to study further the polynomial family P, an interesting collection of polynomials with one complex parameter which arise naturally in the study of Mobius groups. Mathematics can be viewed as consisting of three main fields - algebra, analysis, and geometry - together with many other areas which are developments of the basic ideas in one or more of these fields. Thus algebra is the field which has developed from studying polynomial equations, analysis the field which has its roots in the calculus, and geometry the field that has developed from the ideas of Euclid and the Greeks. It is particularly interesting and satisfying for mathematicians when ideas which belong to one of these fields can be used to solve problems in another. This project concerns some problems which are situated at a crossroads where all three of these fields meet. In particular, one problem is to determine the smallest possible volume of certain g eometric objects using the theory of complex iteration and Mandelbrot sets of analysis together with a polynomial family which arose from algebra. Work in such a part of mathematics can be quite fruitful since the solution of a crossroads problem usually has implications in all of the fields concerned.