This proposal consists of two main sections. The first section deals with coarse geometry and geometric group theory. The PI (together with D. Fisher and K. Whyte) has recently developed a new technique, "coarse differentiation", which can be viewed as a sort of differentiability substitute for quasi-isometries. Of course, conventional derivatives do not make sense for such maps, since they are not even defined on small scales; instead we must go to larger and larger scales. Using this technique, we were able to resolve three longstanding open problems in the field, namely proving the quasi-isometric rigidity of the three-dimensional solvable group Sol, exhibiting a transitive graph which is not quasi-isometric to any Cayley graph, and showing that the two state and the three state lamplighter groups are not quasi-isometric. We list some other potential applications of the method, many of which are to problems which seemed completely out of reach a year ago.
The second section concerns the interrelated analytic study of billiards in rational polygons, moduli spaces of abelian and quadratic differentials, and the dynamics of the SL(2,R) action on these moduli spaces. The PI also proposes to study related questions about the geometry of these spaces, such as their volumes and their SL(2,R) invariant submanifolds. In particular, the PI has found by numerical experiment some polygons which seem to have competely unexpected properties, and proposes to study them further.
Some of the coarse geometry in the the first part of the proposal has unexpected connections to computer science, in particular the existence of efficient algorithms for finding ways to disconnect a graph by cutting as few edges as possible. In fact, one of our proposed problems is taken from this field.
The rational billiard, which is the main subject of study in the second part of the proposal is important in particular for the following reason: Some natural phenomena are "chaotic" (i.e. unpredictable). These are often studied by statistical methods. Others are "integrable" (i.e. predictable and regular). Other phenomena fit somewhere in between. The polygonal billiard is one of the simplest known models of intermediate behavior. As such it has been studied extensively in physics as well, in particular in connection to "quantum chaos".
