Cannon--Thurston maps and the curve complex.
Christopher J. Leininger, University of Illinois at Urbana-Champaign

J. Cannon and W. Thurston proved that a closed hyperbolic
3-manifold M that fibers over the circle with fiber S gives rise to a
sphere-filling curve. This is an equivariant map, now known as a
Cannon--Thurston map, is from the boundary of the universal cover of S to
the boundary of the universal cover of M. Mahan Mitra later extended this
result to any Gromov hyperbolic S--bundle.

The universal S--bundle is not hyperbolic. However the universal
extension does act on the fibration of curve complexes. Masur and Minsky
proved that the complex of curves is Gromov hyperbolic, and the fibration
was studied in joint work with Richard Kent IV and Saul Schleimer.

I'll discuss joint work with Mahan Mj and Saul Schleimer in which we
construct a "universal Cannon--Thurston map". This map is not defined
on the entire Gromov boundary of the universal cover S (by necessity), but
on its domain of definition, is universal. I'll explain this and the
implications.

Colloquium, Department of Mathematics, Rice University
January 10, 2008, 4:00-5:00 PM, HB 227


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