# Mathematics Colloquia and Seminars

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### Boundaries of Teichmuller spaces and end-invariants for hyperbolic 3-manifolds.

**Geometry/Topology**

Speaker: | Jeff Brock, Stanford University |

Location: | 693 Kerr |

Start time: | Wed, May 5 1999, 4:10PM |

In two celebrated boundaries for Teichmuller space due to Bers and Thurston geodesic laminations arise in natural ways:An important conjecture of Thurston's is that E is an injection. As a related issue, we will consider continuity properties of the mapping E. We will show that E is discontinuous, but lower-semi continuous in the quotient topologies. We formulate a topology (the end-invariant topology) on the range in which E is continuous. This topology predicts new data about limiting end-invariants and reveals some fundamental obstructions to developing a complete picture of how they vary.

- A point M in Bers' boundary, a hyperbolic 3-manifold has an associated geodesic lamination that has been "pinched."
- A point L in Thurston's, a measured lamination up to scale, records asymptotic stretching of divergent marked Riemann surfaces. Such geodesic laminations provide a natural mapping from a quotient of Bers boundary to a quotient of Thurston's, by assigning to M its "end-invariant" E(M), the pinched geodesic lamination for M.