Hyperbolic Manifolds: Geometry, Topology, Group Theory and Arithmetic

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Motivated by considerations in topology, geometry, arithmetic andgroup theory the proposal intends to study real, complex andquaternionic hyperbolic manifolds. This will involve the study ofdiscrete groups, and their connections with other research active areas; forexample number theory, and the theory of expandinggraphs. We will also explore various interconnections between thesegeometric objects. This has been fruitful in the past, since previous work has shown how an understanding of higher dimensionalhyperbolic manifolds can impact the topology of 3-dimensional manifolds.The proposal seeks to further understand basic objects in moderngeometry and topology, namely manifolds that admit a hyperbolic metricof some type. Many of the problems in the proposal arecross-disciplinary, and are amenable to study using a broad spectrumof mathematical technqiues; for example from number theory, geometry,topology and group theory.By their nature, progress and solutions to problems in the proposalwill have a broader impact. The problems suggested have applicationsin fields as diverse as number theory (class numbers) and cosmology(and its relation to the topology of 4-dimensional hyperbolicmanifolds) and computer science (expanding graphs). All of these havethe potential to be fertile grounds for the education of graduatestudents.