Program

The fields of Kleinian groups and Teichmüller theory have each seen dramatic changes in recent years. Many new techniques have been developed, major conjectures have been solved, and new directions and connections have been forged. Yet to a large extent progress has been made in parallel without the level of direct communication across these two fields that is clearly warranted. The MSRI program in Teichmüller theory and Kleinian groups will address the need to strengthen connections between these two fields, and reassess new directions for each at a critical time in its history. The recent solutions of the tameness conjecture, density conjecture and the ending lamination conjecture put the study of hyperbolic 3-manifolds and Kleinian groups at a transitional point. Information about the mapping class group and the curve complex that has arisen out of this is already bearing in important ways on questions in Teichmüller theory and the dynamics and geometry of the mapping class group, and the geometric component of the ending lamination conjecture suggests the possibility of effective models and bounds for closed hyperbolic manifolds. Such developments are playing an important role in strengthening our understanding of parameter spaces of Kleinian groups, including their local and global topology. Likewise, an important development in Riemann surface theory has been the discovery of fertile connections between rational billiards, translation surfaces and flows on Teichmüller space and moduli space. A major focus of the program will be to explore this subject and its connections to hyperbolic geometry, and the combinatorics of the complex of curves on a surface. More generally, there have been recent breakthroughs in understanding the extent of the analogies between the mapping class group and Kleinian groups, and the connections to Veech surfaces and the geometry of the mapping class group makes this area one of particular intererst for researchers in flows on moduli space and in hyperbolic geometry alike. We expect the program to serve as a proving ground for the extent of the connections and analogies between these two areas, and we hope it will help start new threads of inquiry to be pursued during the program and after. "A pleated surface on the boundary of the convex core"