Teichmuller theory and Kleinian groups
MSRI, August 16-17 2007

Thursday August 16Friday August 17
9:30-10:30 Ania Lenzhen: Teichmuller space and its metrics Genevieve Walsh: Surfaces in 3-manifolds
11:00-12:00Moon Duchin: The curve complex and its relatives Jane Gilman: A survey of Schottky groups
12:00-2:00Lunch and conversation
2:00-2:30Elmas Irmak: Mapping class groups, curve and arc complexes on surfaces Pallavi Dani: Measuring sets in infinite groups
2:40-3:10Corinna Ulcigrai: Ergodic properties of flows on surfaces Kelly Delp: Convex projective structures on 2-orbifolds
3:10-4:00Coffee and conversation
4:00-4:30Ege Fujikawa: Teichmuller space and moduli space for Riemann surfaces of infinite type Alexandra Pettet: Teichmuller spaces of n-tori
4:40-5:10Sarah Koch: Teichmuller theory and endomorphisms of Pn Asli Yaman: Kleinian groups from the geometric group theory viewpoint


Basic Notions

  • [Duchin]
    The curve complex, introduced by Bill Harvey circa 1978, is one of a collection of combinatorial gadgets associated to surfaces that serve as useful coarse-geometry tools for the study of Teichmuller space and the mapping class group. I'll give a general introduction themed around three main properties: the curve complex is connected, has infinite diameter, and is delta-hyperbolic. A really nice reference for this material is the as-yet-unpublished notes of Saul Schleimer.

  • [Gilman]
    One of the major achievements of nineteenth and twentieth century mathematics was the classification of Riemann surfaces and the establishment of the theory of moduli. These developments had con- sequences in many fields of mathematics including algebraic geometry, combinatorial and Kleinian group theory, complex analysis and Teichmuller theory, quasi-conformal mappings, topology and hyperbolic geometry. The theory of Schottky groups is a recurrent theme in these areas. In this talk we begin with Schottky's 1887 paper and survey subsequent results up to and including the present touching upon results of Vicki Chuckrow, Albert Marden, Bernard Maskit, Hiro-O Yamamoto, H. Sato, Linda Keen, Caroline Series, John Parker, Alan Beardon, Jane Gilman, David Wright, William Goldman, Peter Doyle, Makoto Sakuma, Ser Tan, and Mario Bonk-Bruce Kleiner-and-Sergei Merenkov among others. We conclude with some open problems.

  • [Lenzhen]
    I would like to give some background on geometry of Teichmuller space. I will mainly talk about properties of the Teichmuller, and if time permits, the Weil-Petersson, metrics. In particular, I will discuss the behavior of geodesic rays with respect to the Thurston compactification of Teichmuller space.

  • [Walsh]
    I'll give some background as to how surfaces in 3-manifolds are useful. I'll also survey some recent results and open problems regarding surfaces in 3-manifolds, particularly the problem of virtual fibering, and how these relate to commensurability.

    Research Themes

  • [Dani]
    Consider a finitely generated infinite group G and a subset X. (For example, X may be a subgroup or the set of elements with a particular property.) It is natural to ask: what is the probability that a random element of G lies in X? One way to formalize this is via the asymptotic density of X. This is defined by fixing a word metric, and considering the limit, as r tends to infinity, of the proportion of elements in the ball of radius r belonging to X. I will describe the role of geometry in computing asymptotic densities in some examples and talk about some questions of this nature in the realm of hyperbolic groups and mapping class groups.

  • [Delp]
    Given a convex real projective structure on a two dimensional orbifold, we may equip the universal cover with a Hilbert metric. This metric induces a metric on the base orbifold. In our talk we will talk about recent results in understanding the deformation space of such metrics.

  • [Fujikawa]
    To consider the structure of the moduli space for a Riemann surface of infinite type, we introduce a new space, which is called the intermediate Teichmuller space, between the Teichmuller space and the moduli space. The intermediate Teichmuller space lies also between the Teichmuller space and the asymptotic Teichmuller space. Then we investigate a complex structure of the intermediate Teichmuller spaces as well as the metric structure. Moreover, we determine the biholomorphic automorphism group of the intermediate Teichmuller space.

  • [Irmak]
    I will be talking about mapping class groups of compact, connected, orientable surfaces and some known results about their relation to the automorphism groups of several curve complexes on surfaces. I will also talk about my joint work with J.D. McCarthy. In this work we prove that every injective simplicial map of the arc complex of a compact, connected, orientable surface is induced by a homeomorphism of the surface, and the automorphism group of the arc complex is naturally isomorphic to the quotient of the extended mapping class group of the surface by its center.

  • [Koch]
    Inspired by Thurston's theorem of the characterization of rational maps, J. H. Hubbard posed the "twisted rabbits" problem. This problem was recently solved by L. Bartholdi and V. Nekrashevych using original techniques involving iterated monodromy groups. A key part of their solution contains the construction of a map on a certain moduli space. We discuss Thurston's theorem and present the "twisted rabbits" problem. We then generalize the construction of these Bartholdi-Nekrashevych maps and discuss their significance.

  • [Pettet]
    Let T_n be the Teichmueller space of flat metrics on the n-dimensional torus and identify SL(n,Z) with the corresponding mapping class group. We describe some interesting properties of Ash's spine, which consists of those points at which the systoles generate a finite index subgroup of the fundamental group of the torus.

  • [Ulcigrai]
    We consider area preserving flows on surfaces and investigate their ergodic properties. We will first explain how such surface flows can be described by suspension flows over interval exchange transformations. We will then focus on mixing and weak mixing in flows which are locally Hamiltonian. One of the tools that we use is a renormalization algorithm for interval exchange maps which is very helpful in Teichmuller dynamics.

  • [Yaman]
    Since the 1980s, hyperbolic geometry and geometric group theory have interacted fundamentally, leading to creativity and animation in both areas. One of the main areas of intersection of hyperbolic geometry and geometric group theory is Kleinian groups, discrete groups of isometries of hyperbolic n-space. In this talk, we will define relatively hyperbolic groups and convergence groups, both of which found their motivations and examples in Kleinian groups. Relatively hyperbolic groups were intoduced by Gromov, and give a generalisation into the combinatorial setting of Kleinian groups acting on a hyperbolic n-space with a finite volume quotient, while convergence groups axiomatise the dynamics induced by a Kleinian group on the boundary of hyperbolic n-space.

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