Hyperbolic geometry, topology and dynamics

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AbstractAward: DMS-1005973Principal Investigator: Yair MinskyThe PI proposes to investigate a number of different aspects of hyperbolic geometry in 2 and 3 dimensions, the structure of the representation spaces that parametrize these geometric systems, and the dynamics of the groups that act on them. The SL(2,C)-character variety, X(F), of a group F parametrizes conjugacy classes of representations of F into SL(2,C), the isometry group of hyperbolic 3-space. Discrete faithful elements of X(F) correspond to hyperbolic 3-manifolds, and the PI will work to understand better the interaction between topological and geometric features of these manifolds, as well as the structure of the discrete-faithful locus itself. In addition, he will pursue a study of X(F) as a whole, considered as a dynamical system under the action of the outer automorphism group of F (particularly when F is a free group). A new dynamical decomposition of X(F) was recently discovered, whose structure seems to be rich and relatively unexplored. In addition the PI will study geometric aspects of Teichmuller spaces, which parametrize hyperbolic structures in two dimensions, and of Mapping Class Groups, their natural automorphism groups.The interactions between geometry, topology and dynamics play a entral role in mathematics as well as its applications. A geometric space, such as our own universe or the configuration space of some system, may admit dynamical phenomena such as flows, iterations or group actions. The behavior of these phenomena, as well as the geometry of the space, can be strongly influenced by its topological structure, namely the underlying connective tissue on which the geometry is overlaid. Furthermore, we often find that geometry and dynamics persist at higher levels of abstraction: the collection of all geometric structures on a given space can itself be organized into a new "higher" space, with its own geometry and its own inherent symmetries which give rise to dynamical structure. The interaction between these phenomena at different levels can enrich our insight about the original systems. The PI's own research focuses on particular instances of this general template, namely the geometry and topology of 2- and 3- dimensional spaces, and the corresponding dynamics for their higher parameter spaces. This low-dimensional setting is particularly amenable to our intuition and is partly motivated by direct visual analogies with our physical world, but it also happens, for a variety of reasons, to be a meeting place for a number of different areas of mathematics as well as physics, so that a fuller understanding in this domain can enrich, by analogy as well as direct mathematical connection, our approaches to other parts of mathematics. This award is to be funded jointly by the programs in Topology, Geometric Analysis, and Analysis.