Topology, geometry and arithmetic of hyperbolic 3-manifolds

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Culler and Shalen will continue their research on hyperbolic 3-manifolds. One of the main themes of their work is the connection between topologically defined invariants of such manifolds and their quantitative geometric invariants such as volume. This has involved interactions between very classical techniques in 3-manifold topology, some of which go back to Papakyriakopoulos's work in the 1950's, and more geometric methods such as the log(2k-1)-theorem of Anderson, Canary, Culler and Shalen, the work of Kojima and Miyamoto on hyperbolic manifolds with totally geodesic boundary, and the work of Agol, Dunfield, Storm and Thurston based on properties of the Ricci flow with surgeries. A second theme, which recently has grown out of the first, is the connection between the number-theoretic invariants of a manifold such as its trace field and quantitative geometric invariants such as its Margulis number. This aspect of the work depends on combining the earlier work with new group-theoretic observations, and has already brought into play such deep number-theoretic ingredients as the work of Siegel and Mahler on the unit equation in algebraic number fields.Hyperbolic manifolds are geometric objects that arise in many branches of mathematics and in many applications of mathematics. The first hyperbolic manifold, called hyperbolic space, was discovered in the 19th century and settled---in the negative---the ancient problem of whether Euclid's fifth postulate could be deduced from his other postulates. Hyperbolic manifolds may be thought of as geometric objects which at small scales are indistinguishable from hyperbolic space, but whose large-scale behavior is more complicated. A major theme in modern geometry is the interaction between the quantitative properties of a geometric object, for example those defined in terms of distances, lengths, areas and volumes, and their "topological"properties which are more qualitative and are unchanged when the object is deformed. In the case of hyperbolic manifolds, so much progress has been made in recent years in relating the quantitative and topological theories that they may be said to be completely unified at an abstract level. The present project involves making our understanding the connection in a more concrete way. Doing this turns out to involve deep ideas from many branches of mathematics.