Hyperbolic Geometry Seminar 2003-2004

In 2003-2004, the seminar was organised by Choi Young Eun.

13th July

Kasra Rafi (Univ of California, Santa Barbara, USA)

Geometry of Teichmuller space and the complex of curves

Abstract: Using techniques developed by Masur and Minsky, we study geodesics in Teichmuller space. We give a characterization of short curves along a geodesic and give an estimate for the shortest length of a curve. As a consequence we provide a combinatorial formula for the distance between two points in Teichmuller space.

3rd June

Javier Aramayona (Southampton)

The Weil-Petersson geometry of the five-times punctured sphere

27th May

Reza Chamanara (Indiana)

Simultaneous bending of projectively convex piecewise circular Jordan arcs (CONTINUED)

20th May

Reza Chamanara (Indiana University, USA)

Simultaneous bending of projectively convex piecewise circular Jordan arcs

13th May

Mary Rees (Liverpool)

The geometric model and coarse Lipschitz equivalence direct from Teichmuller geodesics

Abstract: We shall explain a proof of the Ending Lamination Conjecture which uses Teichmuller geodesics directly, restricted, for simplicity, to the case when the ending laminations data is a pair of minimal laminations.

29th April

Caroline Series (Warwick)

Thurston's bending measure conjecture for once punctured torus groups

Abstract: We prove Thurston's bending measure conjecture for quasifuchsian once punctured torus groups. The conjecture states that the bending measures of the two components of the convex hull boundary uniquely determine the group.

25th March

Cliff Earle (Cornell University)

A dynamical approach to the conformal barycenter

Abstract: The conformal barycenter of a probability measure on the unit circle is a central ingredient in Douady and Earle's theory of the barycentric extension of circle homeomorphisms. Recently, Abikoff developed an iteration scheme, called the MAY iterator, for calculating the conformal barycenter. New work by Abikoff, Mitra, and me uses the MAY iterator to provide a new definition of the conformal barycenter. This definition applies to a wider class of probability measures and allows the extension of continuous degree one monotone maps of the circle that are not necessarilty homeomorphisms.

18th March

Al Marden (University of Minnesota)

On PSL(2,C)-representation varieties for Kleinian groups

Abstract: I will give a survey of some results centered on the represention variety of a geometrically finite kleinian group. The most detailed information available is for the special case of a fuchsian surface group. For this case the principal tool to be discussed is the covering of the representation variety by the bundle of complex projective structures, in particular, grafting. The method of complex scaling offers another approach. I will give McMullen's proof of a prototypical case of ``bumping'' of components of the discreteness locus of the representation variety. Apart from this proof the talk will be entirely expository.

11th March

Cyril Lecuire (Warwick)

Bending laminations and convergence of metrics

Abstract: In this talk I will conclude the proof of the main theorem of the first talk which gave necessary conditions for a geodesic measured lamination to be the bending lamination of some convex cocompact hyperbolic metric.

4th March

Cyril Lecuire (Warwick)

Bending laminations and algebraic convergence

Abstract: In this third talk, I will show that if we have a sequence of faithful and discrete representations with converging bending measures (with some restrictions on the limit) then there is a subsequence that converges algebraically.

26th February

Cyril Lecuire (Warwick)

Convex cores of hyperbolic 3-manifolds and bending measured geodesic laminations (continued)

19th February

Cyril Lecuire (Warwick)

Convex cores of hyperbolic 3-manifolds and bending measured geodesic laminations

Abstract: This is the first talk in a series of four. I will give some basic definitions, especially that of a convex core, bending measured laminations and characteristic submanifolds of hyperbolic 3-manifolds. I will then explain some properties of the bending geodesic measured laminations.

5th Februrary

Emmanuel Dufraine (Warwick)

Actions of Surface Groups on Real Trees -- Skora's Theorem (Otal's book, chapter 8)

29th January

Kentaro Ito (Nagoya University, Japan)

Bumping of components of quasi-fuchsian projective structures

Abstract: Let S be a closed hyperbolic surface. Let P(S) be the space of marked projective structures on S and let Q(S) be the subset of P(S) consisting of projective structures with quasi-fuchsian holonomy. It is known that Q(S) has infinitely many components: there is only one component called "standard" and all other components are called "exotic".

In this talk, we explain that each exotic component bumps into the standard component. Moreover we show that each exotic component self-bumps. As a consequence, we can also show that any two components bump. One of the main tools used here is the grafting operation on a projective surface.

22nd January

Caroline Series (Warwick)

Chapter 6 of Otal's book "The Hyperbolisation Theorem for Fibred 3-Manifolds"

15th January

Caroline Series (Warwick)

Chapter 6 of Otal's book "The Hyperbolisation Theorem for Fibred 3-Manifolds"

8th January

Akira Ushijima (Warwick, Kanazawa University)

Volumes of Hyperbolic Tetrahedra

From October to November

Reading seminar of "The hyperbolization theorem for fibered 3-manifolds" by Jean-Pierre Otal

Co-organized with Emmanuel Dufraine.

Current year of the seminar.