| Abstracts of Talks |
Jeff Brock (Brown University): Ending Laminations for Weil-Petersson Geodesics Abstract: The ending lamination for a Weil-Petersson geodesic ray encodes aspects of its asymptotic behavior. In particular, its existence demonstrates a coherence to the set of accumulations on the boundary of the curve complex of the collection of simple closed curves whose length functions have minima below a fixed constant along the geodesic. In other incarnations, such ending laminations give a complete picture of asymptotic geometry. I will describe what is known and unknown about related conjectures for Weil-Petersson geodesics. This is joint work with Howard Masur and Yair Minsky. |
Dick Canary (University of Michigan): A Pressure Metric for the Hitchin Component
Abstract: We construct an analytic Riemannian metric on the
Hitchin component of the space of representations of a closed
surface group into PSL(n,R) which restricts to the Weil-Petersson
metric on the Fuchsian locus. The same techniques produce metrics on
deformation spaces of convex cocompact representations of a
hyperbolic group into PSL(2,C). We also show that the Hausdorff
dimension of the limit set varies analytically over deformation
spaces of convex cocompact representations into any rank one
semi-simple Lie group. This talk describes joint work with Martin
Bridgeman, Francois Labourie and Andres Sambarino.
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Moon Duchin (Tufts University): Typical Teichmüller Geodesics Abstract: I'll review some of the features and pathologies of geodesics in the Teichmüller metric, including the phenomena that are obstructions to hyperbolicity. Then I'll discuss recent work with Dowdall and Masur in which we work out properties enjoyed by generic geodesics, concluding that these obstructions are quantifiably rare, and Teichmüller space is in this sense statistically hyperbolic. |
David Futer (Temple University): Cusp Geometry of Fibered 3-Manifolds Abstract: Let F be a surface with punctures, and suppose that φ: F → F is a pseudo-Anosov homeomorphism fixing a puncture p of F. Then the mapping torus of φ is a hyperbolic 3-manifold Mφ, which contains a maximal cusp corresponding to the puncture p. We show that the geometry of the maximal cusp can be predicted, up to explicit multiplicative error, by the action of φ on the complex of essential arcs in the surface F. This result is motivated by an analogous theorem of Brock, which predicts the volume of Mφ in terms of the action of φ on the pants graph of F. However, in contrast with Brock’s theorem, our result gives effective estimates, and is proved using completely elementary methods. This is joint work with Saul Schleimer. |
Jeremy Kahn (Brown University): The Good Pants Homology and the Ehrenpreis Conjecture Abstract: The Ehrenpreis conjecture states that given any two closed Riemann surfaces S and T of genus greater than 1, and any K > 1, there are finite covers S and T of the two surfaces, and a K-quasiconformal map between them. In joint work with Vladimir Markovic, we prove the Ehrenpreis conjecture by showing that any closed hyperbolic surface S has a cover S that is made of "good pants" that have been assembled in a good way, and then show that any two "good panted surfaces" have common covers that are close in the Teichmüller metric. We find these good panted covers by showing that the set of good pants (which have boundary lengths close to a given large R) is evenly distributed around every good geodesic. We can assemble a formal sum of good pants to form a good panted surface provided that the formal sum is balanced-that there are an equal number of pants on the two sides of every good geodesic. We define the "good pants homology" to analyze the obstruction to correcting an imbalance, and we show that the good pants homology is equal to the standard homology, which implies that the obstruction is trivial. |
Yair Minsky (Yale University): Squeeze Maps on Hyperbolic Surfaces Abstract: Deformations in Teichmüller space which contract every length function are closely related to three-dimensional affine representations of free groups. The space of such deformations is a convex cone in the tangent space of Teichmuller space. We study the structure of this cone and its boundary. Our most complete description is still only in the lowest dimensional cases, but we present some ideas and questions about the general case. Joint work with Goldman and Margulis. |
Igor Rivin (Temple University): Generic Phenomena in (Mapping Class and Otherwise) Groups -- Many Questions and a Few Answers Abstract: I will talk about the recent results on generic and otherwise subgroups of geometrically interesting groups, and the sad realization that we know a lot less about them than we thought. |
Daniel Wise (McGill University): A Problem About Generic Curves in Surfaces Abstract: Let S be a finite type hyperbolic surface. My aim is to formulate a problem about the behavior of generic closed geodesics in S. I am interested in the problem because it is connected to basic issues in combinatorial group theory. As will be explained, a positive resolution of the problem has applications towards the coherence and local quasiconvexity of one-relator groups.
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Scott Wolpert (University of Maryland): Products of Twists, Geodesic-Lengths and Thurston Shears Abstract: Our goal is to generalize formulas for Fenchel-Nielsen twists, geodesic-lengths and the Weil-Petersson metric to the setting of punctured surfaces triangulated by ideal geodesics. We recall Bonahon’s embedding of Teichmueller space for compact surfaces into the space of transverse cocycles on a maximal geodesic lamination. We further recall Penner’s lambda-length embedding of decorated Teichmuller space for punctured surfaces to the positive Euclidean orthant. For punctured surfaces triangulated by ideal geodesics, we consider weighted sums of geodesics with weights summing to zero at each cusp. We analyze such configurations by “doubling across cusps” and “opening nodes” to obtain compact surfaces with FN twists We show that the dual of a Thurston shear is the total length, that the WP symplectic pairing is given by weight summation by parts at punctures and by a new formula for the Penner form and we present the formula for the WP metric pairing. |