CAREER: Large Scale Geometry and Dynamics in Group Theory

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DMS-0349290Kevin M. WhyteThe program of studying discrete groups via the quasi-isometric geometry of their Cayley graphs was suggested by Gromov in his address at the 1982 International Congress of Mathematicians. This program has two complementary parts: finding invariants to show that different groups are not quasi-isometric, and constructing quasi-isometries to show that similar groups are quasi-isometric. The PI has made several contributions to this program in the past, including a recent proof of quasi-isometric rigidity for the group of integral affine transformations and the group of automorphisms of a surface group (the latter in joint work with Lee Mosher). The PI proposes significant extensions of these results in the context of the development of the subject of large scale dynamics. In addition to rigidity phenomena, the proposed research includes a detailed study of less rigid groups, which should help to clarify questions about quasi-isometric invariance of many algebraic and combinatorial properties of groups. Roughly speaking, large scale (or coarse) geometry is the study of geometric properties of objects "seen from far away". From this perspective, any bounded object is indistinguishable from a point, and a line of dots is indistinguishable from a solid line. This sort of geometry has been influential recently in many areas of mathematics, notably group theory, topology, and geometric analysis. This research will explore the large scale geometry of several classes of mathematical objects, both classical geometric spaces and objects only now being viewed in a geometric manner. There has been a tremendous amount of interest in this area of study recently, with many of the experts in the field located at one of the universities in the Chicago area. The proposal includes a variety of programs to develop interaction between mathematicians, and especially their graduate students, within this community.