Coarse Differentiation and Teichmuller Dynamics

University Of Chicago, Chicago IL

Investigators

Abstract

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 lamplighter groups Z wreath product Z mod 2 and Z wreath product Z mod 3 are not quasi-isometric. We list some recent progress and other potential applications of the method, many of which are to problems which seemed completely out of reach before. 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 and the geodesic flow on these moduli spaces. In recent work with M. Mirzakhani, the PI was able to resolve a twenty year old conjecture by V. Veech in this area. Some of the techniques are based on a loose analogy with flows on locally symmetric spaces. Even though the moduli spaces of differentials are substantially different, the PI was and is involved in transferring some of the symmetric space techniques to this setting. We propose additional research in this direction. 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, some of our ideas were already used to solve problems in this field. 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 system, which is one of our main subjects of study in the second section of the proposal, is a good model of intermediate behavior. As such it has been studied extensively in physics as well, in particular in connection to "quantum chaos".