Hyperbolic geometry and 3-manifolds

Summary

My theme is that geometrical methods yield more information about manifolds than do purely topological methods and in dimension three geometric methods are often applicable, A good example of this is the recent proof of the Smith Conjecture.

Here is the strongest possible conjecture asserting that one can always use geometry when studying 3-manifolds. The Poincaré Conjecture is a very special case.

CONJECTURE, Every compact 3-manifold M with incompressible boundary has a canonical decomposition into geometric pieces, i.e. by cutting M along a canonical family of disjoint, 2-sided, closed surfaces each homeomorphic to S², P², T²or the Klein bottle, one can obtain geometric pieces.

When I say that a 3-manifold M is geometric, I mean that the interior of M has a complete geometric structure modelled on some homogeneous space. By a homogeneous space, I shall mean a space X and a transitive group G of homeomorphisms of X with the property that G_X, the stabiliser of x, is compact for every x in X. It follows that X admits a G-invariant metric. We will always assume that X is equipped with such a metric and we will usually assume that G is maximal i.e. the full isometry group of X.

A manifold M without boundary has a (X,G)-structure if it is locally homeomorphic to open subsets of X and there is an atlas of charts such that all the overlap maps lie in G.