9305852 Gehring This project is concerned with mathematical research on geometric problems associated with classical complex analysis and its more modern derivatives of quasiconformal mapping and Mobius groups in two and higher dimensions. The work on groups focuses on questions related to the geometry of discrete two generator groups with an elliptic generator. It is possible to study the groups through triples of complex numbers derived from the traces of the generators. But not all triples come from two-generator groups and efforts to determine description of the sets which determine discrete groups or the sets arising from elliptic generators of order greater than three will be undertaken. Applications of this work are made to the study of three- dimensional complete hyperbolic orbifolds. In particular substantial improvements in known lower bounds for the volume of these orbifolds has been found. The primary goal of this project is to show that the volume of every hyperbolic orbifold is the same as those for which the corresponding group contains an elliptic element of order greater than three. This value is approximately .03905... Work in quasiconformal mapping continues on the basic issue of determining when a domain in dimension greater than two is quaiconformally equivalent to the unit ball of the same dimension. Sufficient conditions are known through Schoenflies type theorems. If necessary conditions are found, they will probably have to be formulated in certain types of k- connectivity of the domains. Finally, students supported by this award will consider how classical function theory relations discovered by Hardy and Littlewood carry over to bounded quasidisks. Studies in geometric function theory focus on how domains are mapped into one another by smooth functions on which various distortion restrictions are placed. Questions of which domains can be transformed into one another arise naturally and this point of view automati cally leads to group theoretic questions of fundamental consequence. ***
