Thin Groups and Dynamics

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A homogeneous space is a geometric object that manifests a continuous family of symmetries. Dynamics of flows on homogeneous spaces are related to many natural problems in number theory and geometry. Traditionally, the homogeneous spaces of interest in most studies have finite volume. However, it has recently been discovered that the dynamics on homogeneous spaces of infinite volume also arise in many natural problems such as Apollonian circle packings and in building efficient networks. This proposal will investigate this rich topic.The proposal deals with dynamics on infinite homogeneous spaces, which turns out to yield quite a rich theory and has several deep and striking applications. The proposed study 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.