Dynamics and Kleinian Groups

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The three-dimensional hyperbolic space can be modeled by the upper half space of the three-dimensional Euclidean space, where the shortest path between two points is either along a vertical line or a vertical semi-circle. The boundary of this space is the plane together with the point at infinity. Most often the trajectory of a collection of symmetries in this space accumulates on a fractal on the plane, the study of which is related to dynamics of flows on the hyperbolic manifold, given by the quotient of the hyperbolic three space by the symmetry group. One beautiful example of such fractals is the Apollonian gasket which is constructed by a theorem proven by the ancient Greek geometer Apollonius. Fractals are abundant in nature as can be seen in the coastlines, ocean waves, Romanesco cauliflowers, etc. A drop of oil in soy sauce creates a circle packing akin to the Apollonian gasket. This project is expected to be of value in interpreting the structure of fractals arising in nature.Whereas most of hyperbolic three manifolds have infinite volume, the dynamical rigidity properties of hyperbolic manifolds have been investigated mostly for finite volume cases. The project concerns with dynamics on infinite volume hyperbolic manifolds, which turn out to be quite a rich theory and has several deep and striking applications. This project involves new interactions between hyperbolic geometry, Kleinian groups, ergodic theory and number theory. Investigating this new territory requires many new ideas both from geometry and dynamics and opens up new interactions between these fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.