Warwick Mathematics Institute

Dehn filling/Cannon-Thurston days

Organized by Saul Schleimer, and Caroline Series.

Cannon-Thurston day: 2009 May 2, 9:30am, room B3.02

Caroline Series (Warwick)

The Cannon-Thurston map: From round circles to space filling curves.

Abstract: The limit set of a Fuchsian group G, corresponding to a hyperbolic closed surface S, is a round circle L. At the other extreme, it is possible to have a representation of G into SL(2,C) for which the limit set is the entire Riemann sphere C^. Such representations are called doubly degenerate. The Cannon-Thurston map (if it exists) is a G-equivariant map from L into C^. In particular therefore, it is a space filling curve.

In this introductory talk I will explain how the C-T map can be understood in terms of the evolution of the limit set and the corresponding 3-manifold as the group G is deformed from a Fuchsian to a quasifuchsian group, through a sequence of cusp groups in which certain closed curves become parabolic, finally arriving at a doubly degenerate group in which a pair of geodesic laminations, called the ending laminations, are `pinched' to zero length.

The talk will be illustrated with many pictures of limit sets. I will discuss some features of the pictures which inspired results about the degeneration to the C-T map due to Scorza, and more recently to myself and Sakuma. I will also explain terms such as geodesic laminations, cusp groups, degenerate groups and ending laminations.

Cannon-Thurston day: 2009 May 2, 10:45pm, room B3.02

Brian Bowditch (Warwick)

Stacks and the Cannon-Thuston map

Abstract: The Cannon-Thurston map can be defined in the fairly general setting of a ``stack'' of Gromov hyperbolic ``sheets'', which are glued together via uniform quasi-isometries and which satisfy a certain quasi-convexiy condition. Such a stack is itself hyperbolic; this follows from work of Bestvina and Feighn. Using ideas of Mahan Mj one can abtain a continuous map from the boundary of any sheet to the boundary of the stack. We will outline the main arguments involved.

Cannon-Thurston day: 2009 May 2, 1:15pm, room B3.02

Makoto Sakuma (Hiroshima)

On punctured torus bundles: Comparing two tessellations on the complex plane

Abstract: (Joint work with Warren Dicks.) To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy, there are associated two tessellations of the complex plane: one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra, and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group. In this talk, we explain the relation between these two tessellations. I would also like to discuss possible generalization of the result.

Cannon-Thurston day: 2009 May 2, 3:00pm, room B3.02

Mahan Mj (Vivekananda)

Model geometries and the Cannon-Thurston map

Abstract: I shall spend some time discussing the proof of the existence of a Cannon-Thurston map for a specific model geometry (called amalgamation geometry). I'll also indicate how a general surface group has a model geometry that is a slight generalization of amalgamation geometry.

Dehn filling day: 2009 May 1, 10:45am, room B3.02

Stefan Friedl (Warwick)

Introduction to Dehn surgery

Abstract: I will introduce the subject of Dehn surgery, recalling all the relevant definitions and summarizing some of the main results in the field.

Dehn filling day: 2009 May 1, 1:15pm, room B3.02

Marc Lackenby (Oxford)

The maximal number of exceptional Dehn fillings

Abstract: I will outline the proof of two old conjectures of Cameron Gordon. The first states that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10. The second states the maximal distance between exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 8. The proof uses a combination of new geometric techniques and rigorous computer-assisted calculations. This is joint work with Rob Meyerhoff.

Dehn filling day: 2009 May 1, 2:30pm, room B3.02

Steve Boyer (UQAM)

Toroidal Dehn filling of hyperbolic 3-manifolds

Abstract: Sharp bounds have been determined for the distance between most kinds of exceptional filling slopes on the boundary of hyperbolic manifolds. The one family of slopes yet to be successfully dealt with are those which yield small Seifert spaces. In this talk I will report on joint work with Cameron Gordon and Xingru Zhang where we obtain good bounds for the distance between a toroidal filling slope and a small Seifert filling slope.

Dehn filling day: 2009 May 1, 4:00pm, room B3.02

Michel Boileau (Toulouse)

Dehn fillings and geometric structures on 3-manifolds

Abstract: The Dehn filling construction is central in the theory of 3-manifolds. In this introductory talk we will explain how this construction can be used to study geometric structures on 3-manifolds.

Information on past talks. This page was last touched Thu Mar 4 22:01:05 GMT 2010