Mathematical Sciences: Problems in Low Dimensional Manifolds

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Long plans to examine several interrelated questions in the theory of low dimensional manifolds and group theory. The specific problems concern general hyperbolic 3-manifolds (whether their fundamental groups can be subgroup separable), the particular case of hyperbolic surface bundles over the circle (same question, and other specific questions related to the behavior of the entropy and volume functions), and cobordism of surface automorphisms (whether there is an algorithm to determine when a lamination is a limit of discs). These problems are related to Long's established interests in several classical problems in the theory of hyperbolic manifolds and knot and surface cobordism. Geometric topological investigations such as these develop methods for analysing qualitative properties of geometric objects, e.g. knottedness. In some contexts understanding these properties is every bit as valuable as understanding rigid metric properties.