Title and Abstracts: FRG Conference 2009

Shinpei Baba, UC Davis and Bonn

Title: Complex projective structures with Schottky holonomy

Abstract: A (complex) projective structure is a certain geometric structure on an orientable surface, and it corresponds to a representation of the fundamental group of the surface into PSL(2,C). On the other hand, such a (fixed) representation may correspond to infinitely many distinct projective structures. William Goldman gave a characterization of projective structures corresponding to an isomorphism from onto a quasifuchsian group, using a surgery operation called ``grafting''. We will give an analogous characterization of projective structures corresponding to an epimorphism from the surface group onto a Schottky group, where the genus of the surface is equal to the rank of the Schottky group.

Ian Biringer, U of Chicago and Yale

Title: Pseudo-Anosov Maps and Handlebodies

Abstract: Let H be a handlebody with boundary S. We will show that the attracting lamination of a pseudo-anosov map f : S -> S is a limit of meridians in PML(S) if and only if some power of f extends to a subcompressionbody of H. The proof is through 3-dimensional hyperbolic geometry. Joint with J. Johnson and Y. Minsky.

Martin Bridgeman, Boston College

Title: The orthospectra of finite volume hyperbolic manifolds with totally geodesic boundary and associated volume identities.

Abstract: Given a finite volume hyperbolic n-manifold M with totally geodesic boundary, an orthogeodesic of M is a geodesic arc which is perpendicular to the boundary. For each dimension n, we show there is a real valued function F_n such that the volume of any M is the sum of values of F_n on the orthospectrum (length of orthogeodesics). For n=2 the function F_2 is the Rogers L-function and the summation identities give dilogarithm identities on the Moduli space of surfaces.

Dave Gabai, Princeton

Title: Ultra large hyperbolic 3-manifolds

Joel Hass, UC Davis

TItle: Harmonic and simplicial harmonic maps of surfaces

Abstract: Harmonic maps have proven to be a useful tool in the study of negatively curved manifolds. We will discuss the idea of a simplicial harmonic map, and some applications. These surfaces, introduced in recent joint work with Peter Scott, combine aspects of the smooth and discrete settings. They give a simple yet powerful method to control the geometry of families of surfaces.

Jeremy Kahn, Stony Brook

Title: Essential immersed surfaces in closed hyperbolic 3-manifolds

Abstract: We prove that fundamental group of a closed hyperbolic 3-manifold contains a surface subgroup. The subroups are quasifuchsian groups 1 + epsilon close to a fuchsian group. We prove this result by showing via mixing of the geodesic flow that randomly determined pairs of pants are sufficiently uniformly distributed to fit together into a closed almost flat surface. This is joint work with Vladimir Markovic.

Steve Kerckhoff, Stanford

Title: Local rigidity of hyperbolic manifolds with geodesic boundary

Abstract. Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n > 3 then the holonomy representation of W into the isometry group of hyperbolic n-space is infinitesimally rigid.

Francois Labourie, Université Paris-Sud

Title: Surfaces groups in non compact groups

We survey in this talk representations of surface groups in a non compact group G whose propertes generalise those of quasi-fuchsian representations. These representations are defined dynamically using classical notions in hyperbolic dynamics. We show that many properties of classical Teichmüller theory extend to this setting, in particulat when G is a split group such as SL(n,R): discreteness, faithfullness, properness of the action of the mapping class group, MCShane identity... We finally explain a conjecture describing these representations in the language of complex analysis and explain partial results.

Chris Leininger, UIUC

Title: Surface homeomorphisms in 3-manifold topology I, II, and III.

Abstracts: I. In this first lecture, I will discuss some of the connections between 3-manifold topology and surface homeomorphisms (e.g. heegaard splittings, gluing problems, and mapping tori). Then I will describe Thurston's classification of mapping classes, give examples, and describe how the type of a mapping class is reflected in 3-manifold topology.

Title: Complex projective structures with Schottky holonomy

Abstract: A (complex) projective structure is a certain geometric structure on an orientable surface, and it corresponds to a representation of the fundamental group of the surface into PSL(2,C). On the other hand, such a (fixed) representation may correspond to infinitely many distinct projective structures. William Goldman gave a characterization of projective structures corresponding to an isomorphism from onto a quasifuchsian group, using a surgery operation called ``grafting''. We will give an analogous characterization of projective structures corresponding to an epimorphism from the surface group onto a Schottky group, where the genus of the surface is equal to the rank of the Schottky group.

Ian Biringer, U of Chicago and Yale

Title: Pseudo-Anosov Maps and Handlebodies

Abstract: Let H be a handlebody with boundary S. We will show that the attracting lamination of a pseudo-anosov map f : S -> S is a limit of meridians in PML(S) if and only if some power of f extends to a subcompressionbody of H. The proof is through 3-dimensional hyperbolic geometry. Joint with J. Johnson and Y. Minsky.

Martin Bridgeman, Boston College

Title: The orthospectra of finite volume hyperbolic manifolds with totally geodesic boundary and associated volume identities.

Abstract: Given a finite volume hyperbolic n-manifold M with totally geodesic boundary, an orthogeodesic of M is a geodesic arc which is perpendicular to the boundary. For each dimension n, we show there is a real valued function F_n such that the volume of any M is the sum of values of F_n on the orthospectrum (length of orthogeodesics). For n=2 the function F_2 is the Rogers L-function and the summation identities give dilogarithm identities on the Moduli space of surfaces.

Dave Gabai, Princeton

Title: Ultra large hyperbolic 3-manifolds

Joel Hass, UC Davis

TItle: Harmonic and simplicial harmonic maps of surfaces

Abstract: Harmonic maps have proven to be a useful tool in the study of negatively curved manifolds. We will discuss the idea of a simplicial harmonic map, and some applications. These surfaces, introduced in recent joint work with Peter Scott, combine aspects of the smooth and discrete settings. They give a simple yet powerful method to control the geometry of families of surfaces.

Jeremy Kahn, Stony Brook

Title: Essential immersed surfaces in closed hyperbolic 3-manifolds

Abstract: We prove that fundamental group of a closed hyperbolic 3-manifold contains a surface subgroup. The subroups are quasifuchsian groups 1 + epsilon close to a fuchsian group. We prove this result by showing via mixing of the geodesic flow that randomly determined pairs of pants are sufficiently uniformly distributed to fit together into a closed almost flat surface. This is joint work with Vladimir Markovic.

Steve Kerckhoff, Stanford

Title: Local rigidity of hyperbolic manifolds with geodesic boundary

Abstract. Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n > 3 then the holonomy representation of W into the isometry group of hyperbolic n-space is infinitesimally rigid.

Francois Labourie, Université Paris-Sud

Title: Surfaces groups in non compact groups

We survey in this talk representations of surface groups in a non compact group G whose propertes generalise those of quasi-fuchsian representations. These representations are defined dynamically using classical notions in hyperbolic dynamics. We show that many properties of classical Teichmüller theory extend to this setting, in particulat when G is a split group such as SL(n,R): discreteness, faithfullness, properness of the action of the mapping class group, MCShane identity... We finally explain a conjecture describing these representations in the language of complex analysis and explain partial results.

Chris Leininger, UIUC

Title: Surface homeomorphisms in 3-manifold topology I, II, and III.

Abstracts: I. In this first lecture, I will discuss some of the connections between 3-manifold topology and surface homeomorphisms (e.g. heegaard splittings, gluing problems, and mapping tori). Then I will describe Thurston's classification of mapping classes, give examples, and describe how the type of a mapping class is reflected in 3-manifold topology.

II. I will introduce various spaces associated to a surface S (e.g. the Teichmuller space, Thurston's measured lamination space, curve complex, pants graph, etc.). I will talk about the action of the mapping class group Mod(S) on these spaces, and discuss some important features of each. I will also discuss how the actions of mapping classes can be used to derive information about the geometry/topology of 3-manifolds.

III. In this last lecture, I will focus on the mapping torus of a pseudo-Anosov, and talk about recent joint work with Benson Farb and Dan Margalit. The new features of this work are uniform estimates, as the surfaces vary, for the topological complexity of mapping tori of pseudo-Anosov homeomorphism. We use this to find a characterization of the "small complexity" pseudo-Anosov homeomorphisms.

Aaron Magid, U of Michigan and U of Maryland

Title:
Deformation Spaces of Kleinian Surface Groups are Not Locally Connected

Abstract:
For any closed surface S, the deformation space AH(S) is the space of all marked hyperbolic 3-manifolds homotopy equivalent to S. After reviewing some of the classical results that describe topology of the interior of AH(S), we will show that there are certain points on the boundary where AH(S) is not locally connected. This is a generalization of Ken Brombergs result that the space of Kleinian punctured torus groups is not locally connected.

Jessica Purcell, BYU

Title: Hyperbolic structures on compression bodies

Abstract:
We study Ford domains of geometrically finite structures on compression
bodies obtained from adding a one-handle to a torus cross an interval. We
find that in many cases, the topology of the compression body predicts
geometric information on its geometry. Namely, the core of the attached
one-handle is geodesic. These results have applications to the geometry
of single cusped tunnel number one manifolds. This is joint with Marc
Lackenby.

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