CAREER: Homology and Geometry of Groups

University Of South Alabama, Mobile AL

Investigators

Abstract

Abstract for DMS - 0132514 In the past the investigator developed techniques to study Gromov hyperbolic groups. There are several applications of those techniques in topology, geometry, algebra, analysis. In particular, hyperbolic groups were characterized by bounded cohomology, extending results my M. Gromov. This characterization was used by A. Connes and H. Moscovici for a proof of the Novikov conjecture for hyperbolic groups. S. Gersten provided several descriptions of hyperbolicity in terms of various (co)homology theories, and later the investigator made contributions in that direction. Recently, Guoliang Yu and the investigator were able to prove the Baum-Connes conjecture for hyperbolic groups and their subgroups. The techniques will be further developed to address these and other directions. For example, (co)homological descriptions of relative hyperbolicity will be studied. As another application, the conformal and other geometric structures on hyperbolic groups will be constructed. It is the investigator's hope that this will lead, in particular, to a better understanding of open conjectures in 3-dimensional topology. It is an important and interesting task to investigate shapes of spaces, in particular of the one in which we live. Geometry studies spaces of various shapes. Group theory studies groups, that is, sets of symmetries of spaces. Geometric group theory provides links between spaces and groups. Every group has an orbit (!) in some space (!). The orbit is only a part of the space, but once something is known about the the group, geometric group theory provides information not only about the orbit, but about the whole space as well. Tools from several areas of mathematics are used for that. The research part of this proposal is to develop and use those tools. Another part of the proposal is organizing and supporting the G^3 = "Geometric group theory on the gulf coast conference" that would allow exchange of ideas among mathematicians of various research interests.