### 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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