Averaging Methods in Coarse Geometry

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This proposal consists of two main sections. The first section dealswith coarse geometry and geometric group theory. The PI (together withD. Fisher and K. Whyte) has recently developed a new technique,"coarse differentiation", which can be viewed as a sort ofdifferentiability substitute for quasi-isometries. Of course,conventional derivatives do not make sense for such maps, since theyare not even defined on small scales; instead we must go to larger andlarger scales. Using this technique, we were able to resolve threelongstanding open problems in the field, namely proving thequasi-isometric rigidity of the three-dimensional solvable group Sol,exhibiting a transitive graph which is not quasi-isometric to anyCayley graph, and showing that the two state and the three statelamplighter groups are not quasi-isometric. We list some otherpotential applications of the method, many of which are to problemswhich seemed completely out of reach a year ago.The second section concerns the interrelated analytic study ofbilliards in rational polygons, moduli spaces of abelian and quadraticdifferentials, and the dynamics of the SL(2,R) action on these modulispaces. The PI also proposes to study related questions about thegeometry of these spaces, such as their volumes and their SL(2,R)invariant submanifolds. In particular, the PI has found by numericalexperiment some polygons which seem to have competely unexpectedproperties, and proposes to study them further.Some of the coarse geometry in the the first part of the proposal hasunexpected connections to computer science, in particular theexistence of efficient algorithms for finding ways to disconnect agraph by cutting as few edges as possible. In fact, one of ourproposed problems is taken from this field. The rational billiard, which is the main subject of study in thesecond part of the proposal is important in particular for thefollowing reason: Some natural phenomena are "chaotic"(i.e. unpredictable). These are often studied by statisticalmethods. Others are "integrable" (i.e. predictable and regular). Otherphenomena fit somewhere in between. The polygonal billiard is one ofthe simplest known models of intermediate behavior. As such it hasbeen studied extensively in physics as well, in particular inconnection to "quantum chaos".