Deformations of Geometric Structures and Related Topics

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Modern low-dimensional topology is largely concerned with studying the various kinds of geometry that spaces can possess. Two rather different geometries are familiar to us all, specifically Euclidean plane geometry, where the sum of the angles of a triangle is 180 degrees, and the geometry on a sphere (e.g. the Earth), where the sum of angles of a triangle formed from arcs of great circles is greater than 180 degrees. A third, perhaps less familiar geometry, is hyperbolic or non-Euclidean geometry, discovered in the early nineteenth century; here the angle sum of a triangle is in fact less than 180 degrees. Hyperbolic geometry has since assumed a position of pre-eminence in mathematics, owing to its richness and beauty. It should be noted that the mathematics underlying hyperbolic geometry is also important in physics, in particular with regard to special and general relativity. The investigators will acquire a powerful multiprocessor computer with an unusually large amount of memory, and will implement methods they have devised for analyzing deformations of geometries. Knowing the behavior of a geometry under deformations, and the extent to which it can be deformed before it collapses, is an important factor in understanding its nature. The investigators are particularly interested in projective geometry, and a fascinating variant of hyperbolic geometry known as complex hyperbolic geometry (the relation between the "standard" and complex versions is somewhat analogous to the relation between the real numbers and the complex numbers.) The investigators will need to handle complicated objects, including fractal curves, in 3, 4 or 5 dimensions, necessitating much computer power.The goal of this project is to pursue, with the help of a powerful multiprocessor computer, two avenues of research in low-dimensional topology, each concerned with representations of manifold and orbifold fundamental groups. These two topics share an urgent need for an exceptional amount of computer memory, namely a minimum of 64 GB. The first topic concerns deformations of real hyperbolic structures on 2- and 3-dimensional manifolds and orbifolds to structures arising from real projective or complex hyperbolic geometry. The investigators had previously discovered that certain sporadically occurring closed hyperbolic 3-manifolds with small volume admitted such deformations; however, the underlying cause remains a mystery, and the study of degenerations of the resulting structures is virtually uncharted territory. On the other hand, the existence of complex hyperbolic deformations for the 2-sphere with three cone points (of suitable orders) is well established, but again almost nothing is known about degenerations of these structures. The investigators will use the new computer to implement a battery of methods for establishing discreteness of the associated holonomy representations, including a promising new technique based on the natural fibration from (2n+1)-dimensional real projective space to n-dimensional complex projective space. The second topic of research concerns various conjectures related to representations of 3-manifold groups, including knot groups, onto finite groups.The need to store large group conjugacy and multiplication tables again mandates the availability of unusual amounts of computer memory.