Large-scale geometry of groups and spaces with nonpositive curvature

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Proposal: DMS-9972047PI: Bruce KleinerAbstract: The proposed research is in the area of interaction between group theory, differential geometry, dynamics, and topology which evolved during the last 100 years from the study of surfaces with negative curvature. By thinking of discrete groups -- such as fundamental groups of compact manifolds -- in terms of their Cayley graphs, one can view them as geometric objects in their own right. This viewpoint, which is present throughout the proposal, leads to some remarkable insights (Mostow rigidity, Stallings' theorem, Gromov's theorem on groups of polynomial growth, etc.) and to the striking fact that many discrete groups (for example word hyperbolic groups) act in a canonical way on a ``boundary''. The first part of the proposed research focusses on biLipschitz maps. A byproduct of the project, if it is successful, will be an answer to the question of when certain aperiodic tilings of the plane (for example, Penrose tilings) are weakly combinatorially equivalent to the usual square tiling, in the sense that one can match the tiles of the two tilings so that adjacent tiles are matched with nearby tiles. The second project addresses Solv, the only 3-dimensional Thurston geometry that remains mysterious from a large-scale viewpoint. The remaining topics concern important issues in the fields of nonpositively curved spaces (the dynamics of the geodesic flow and the structure of the fundamental group), hyperbolic groups (the topology of the boundary and its relation to group structure), and Poincare duality groups (the relation between Haken PD(3) groups and 3-manifold groups). The proposal has two main themes: symmetry, classification, and recognition of tiling patterns; and dynamics. Tilings of the plane and 3-space have been studied intensely for centuries, and the resulting theory is fundamental in many branches of mathematics, as well as in solid state physics (crystal structure). In the 1970's, mathematicians discovered striking new applications of tilings (especially tilings of hyperbolic space, the geometry underlying a number of Escher's images), by using the ``rough'' pattern of the tilings rather than the precise shape of the tiles and their intersection pattern. From this new viewpoint two tilings would be considered "roughly the same" if there is a one-to-one correspondence between the tiles of one tiling with the tiles of the other so that adjacent tiles in one tiling correspond to nearby tiles in the other. This is a very active area of research with connections to many areas of mathematics. Several of the proposed research topics relate to this: for example it is an open problem if the tilings associated with quasi-crystal structure are always roughly the same as classical crystal tilings. Another theme in the proposal is geodesic motion -- the motion of a free particle -- in a certain class of spaces. These systems have a long history and provide especially simple examples of ``chaotic'' mechanical systems; they can also be used to model a system of several billiard balls on a billiard table with circular (or convex) bumpers. The objective here is to describe and classify, at least in certain classes of examples, all possible trajectories of the system.