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Length and the Gromov Norm of a 3-manifoldGeometry/Topology
|Speaker:||Chris Jerdonek, Mathematics, UC Davis|
|Start time:||Wed, Nov 17 1999, 4:10PM|
Gromov defined a topological invariant that is proportional to the volume for hyperbolic 3-manifolds. Parallel to this I will introduce a topological measure of the length of an isotopy class of curves in a 3-manifold that is proportional to the length for geodesics in a hyperbolic manifold. This allows a topological characterization of the isotopy class of a geodesic in a hyperbolic 3-manifold. In particular, the length is defined for 3-manifolds that a priori are not known to have hyperbolic structures. This could provide therefore a missing step in the cone-manifold program. It may not be too difficult to verify, for closed manifolds with positive Gromov norm, the basic properties of the length function needed for this step to proceed. And curiously, it is not obvious that the length is not meaningful for manifolds with zero Gromov norm.