3-Manifolds and Geometric Group Theory

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Proposal: DMS-9971555PI: Peter ScottAbstract: The proposer plans to investigate purely algebraic analogs of the theoryof the characteristic submanifold of a Haken 3-manifold. Some results inthis direction have already been obtained by several authors but theiruniqueness results are much weaker than that in the 3-manifold context. Itis expected that such results should also yield new understanding of thepurely topological theory of 3-manifolds. The universe appears to be 3-dimensional and it is a very interestingquestion to understand its shape. There is an abstract theory of spacescalled 3-manifolds which attempts to describe all the possibilities forthe shape of such a space. While this theory is not complete, there is agreat deal known. In particular, many (and perhaps all) such spaces can bedecomposed into simple geometric pieces in a natural way. In thisproposal, it is planned to investigate further the properties of thisdecomposition into geometric pieces.