Geometry, Topology and Dynamics of Character Varieties
(18 Jun - 15 Aug 2010)

Co-sponsored by Global COE (Center of Excellence) of the Tokyo Institute of Technology, Compview
and the National Science Foundation (USA)


~ Abstracts ~

 

On volumes of hyperbolic orbifolds
Ilesanmi Adeboye, University of California, USA


We derive an explicit lower bound for the volume of a hyperbolic n-orbifold. Our estimates follow from H. C. Wang's work on lattices of Lie groups and some geometrical arguments. This is joint work with Guofang Wei.

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Hyperbolic structures on surfaces
Javier Aramayona, National University of Ireland at Galway, Ireland


We will cover some of the basic aspects of the theory of Teichmuller spaces, from the point of view of hyperbolic geometry. Starting with elementary hyperbolic geometry, we will describe the Teichmuller space, construct the Fenchel-Nielsen coordinates, study the action of the mapping class group, and end with a discussion of Thurston's compactification. We will not assume any prior knowledge of the topic.

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Two-sided combinatorial volume bounds for non-obtuse hyperbolic polyhedra
Christopher Atkinson, Temple University, USA


We give a method for computing upper and lower bounds for the volume of a non--obtuse hyperbolic polyhedron in terms of the combinatorics of the $1$--skeleton. We introduce an algorithm that detects the geometric decomposition of good $3$--orbifolds with planar singular locus and underlying manifold $S^3$. We also classify the Coxeter polyhedra which give rise to orbifold Seifert--fibered spaces. The volume bounds are then derived by analyzing the components of the decomposition and applying our previous results that give volume bounds for right--angled hyperbolic polyhedra.

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Invariant subsets of finite volume homogeneous spaces
Yves Benoist, Université Paris-Sud, Orsay, France


Let X be a finite volume quotient of a connected semisimple Lie group G with no compact factor and H be a Zariski dense subgroup of G. We prove that every H-orbit closure is an orbit under a larger group. For that we classify the probability measures on X which are stationary under a probability measure on G whose support is compact and spans H: the ergodic ones are homogeneous under a larger group.
This is joint work with Jean-Francois Quint.

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Extending pseudo-Anosov maps into handlebodies
Ian Biringer, Yale University, USA


Let f:S -> S be a pseudo-Anosov homeomorphism of the boundary of a handlebody. We show how the attracting lamination of f determines whether (a power of) f extends into the handlebody. The proof rests on an analysis of the accumulation points of a certain sequence in the PSL(2,C) character variety of S.

Joint work with Jesse Johnson and Yair Minsky.

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Fixed points of the compositions of earthquakes
Francesco Bonsante, University of Pavia, Italy


Let S be a closed surface of genus at least 2. Given a measured geodesic lamination L on S, the left earthquake along L is a diffeomorphism of the Teichmueller space of S. We will prove that if two laminations fill up the surface, the composition of the corresponding left earthquakes admits a fixed point in the Teichmueller space. In particular, we will show that this result can be rephrased in terms of Anti de Sitter geometry using a remark of J.Mess. The key estimate in the proof is then achieved using the Anti de Sitter formulation.
This is a joint work with J.-M. Schlenker.

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Knot commensurability and the Berge conjecture
Steve Boyer, Université du Québec à Montréal, Canada


In this talk I will report on joint work with Michel Boileau, Radu Cebanu and Genevi¨¨ve Walsh. One of the main results is that the commensurability class of the complement of a hyperbolic knot without hidden symmetries contains at most two other knot complements. If it contains at least one other, then both are fibred of the same genus. We can completely characterize those periodic knots (i.e. there is an axis of symmetry disjoint from the knot) without hidden symmetries whose complement is commensurable with another hyperbolic knot complement. If it is not periodic, any such characterization involves a generalization of the Berge conjecture.

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Surface group representations and Higgs bundles
Steven Bradlow, University of Illinois, UC, USA


We will describe what Higgs bundles are, how they relate to surface group representations, and what can be learned from this relationship. We will give special attention to representations into non-compact real reductive Lie groups, where Higgs bundles provide effective tools for, amongst other things, identifying and counting components of the representation variety, and for studying deformations of the representations.

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Ending laminations for Weil-Petersson geodesics
Jeff Brock, Brown University, USA


The notion of an ending lamination for the end of a hyperbolic 3-manifolds records asymptotic combinatorial behavior of simple closed geodesics in the end. A similar notion for Weil-Petersson geodesic rays proves to be useful for understanding the geometry and dynamics of moduli space with the Weil-Petersson metric. We'll review basic constructions and pose directions for further research.

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The topology of deformation spaces of Kleinian groups
Kenneth Bromberg, University of Utah, USA


In the last decade many of the classical conjectures for Kleinian groups have been solved: Marden's tameness conjecture, the Bers-Sullivan-Thurston density conjecture and Thurston's ending lamination conjecture. This last conjecture gives a complete classification of finitely generated Kleinian groups. However, the classifying map is not a homeomorphism for any natural topology on the space of classifying objects. We will give an overview of what is known about the topology of spaces of Kleinian groups and discuss some open conjectures.

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Polynomial automorphisms of the Fricke characters of free groups
Richard Brown, Johns Hopkins University, USA


For a rank-$n$ free group $F_n$ , it is known that its outer automorphism group $Out(F_n)$ acts on the $SL(2,\mathbb C)$-character variety of $F_n$. When the variety is embedded in its proper affine space, this action is by integer polynomial automorphisms. In this talk, we will discuss some of the algebraic and dynamical properties of this action.

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Dynamics on character varieties of 3-manifold groups
Richard Canary, University of Michigan, USA


If M is a compact 3-manifold, the outer automorphism group $Out(\pi_1(M)) of its fundamental group acts naturally on the ${\rm PSL}_2(\bf C}$-character variety X(M) of its fundamental group. This action restricts to an action on the set AH(M) of characters of discrete faithful representations.

We will survey work on the dynamics of the action of $Out(\pi_1(M))$ on both X(M) and AH(M). Moreover, we discuss the topology of the quotient of AH(M) by the action, which one may interpret as the moduli space of hyperbolic 3-manifolds homotopy equivalent to M up to isometry.

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Mapping class group, Cremona group and Hyperbolic geometry
Serge Cantat, University de Rennes I, France


I will describe analogies between the mapping class group of a higher genus closed surface and the Cremona group of all birational transformations of the plane. Both groups act nicely on a hyperbolic space and these actions share nice properties.

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Affine deformations of the holonomy of a three-holed sphere and other surfaces
Virginie Charette, Université de Sherbrooke, Canada


Let S be a hyperbolic surface whose fundamental group is Schottky, such as the three-holed sphere or the once-punctured torus. Then its fundamental group admits proper representations into the group of affine isometries of Minkowski 3-space, a remarkable fact first proved by Margulis. In joint work with Todd Drumm and Bill Goldman, we proved that the space of such proper representations, for the three-holed sphere, is entirely determined by a measure of signed Lorentzian displacement on the three boundary components. We will discuss this result, along with some work in progress on the other surfaces of Euler characteristic -1.

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Spherical triangles and the two components of the SO(3,R)-character space of the fundamental group of a closed surface of genus 2
Suhyoung Choi, KAIST, Korea


We use geometric techniques to explicitly find the topological structure of the space of SO(3, R)-representations of the fundamental group of a closed surface of genus 2 quotient by the conjugation action of SO(3,R). There are two components of the space. We will describe the topology of each of the two components and describe the corresponding SU(2)-character spaces. For each component, there is a sixteen to one branch-covering and the branch locus is a union of 2-spheres and 2-tori.

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On the Chabauty topology on the space of subgroups of a compact group
Yves Cornulier, CNRS, University of Rennes, France


If G is a compact group, the set of its closed subgroups is a compact space for the Hausdorff (or Chabauty) topology. We show that the connected components of this space coincide with the G_0-orbits. This extends a result of S.Fisher and Gartside.

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Deformation space of strictly convex real projective structures on small orbifolds
Kelly Delp, Buffalo State College, USA


A strictly convex real projective manifold, or orbifold, comes equipped with a projectively invariant metric called the Hilbert metric. Let S be a small, orientable, hyperbolic 2-orbifold. That is, S is a two-sphere with exactly three orbifold points, of orders {p, q, r} such that 1/p+1/q+1/r < 1. Goldman and Choi show that the deformation space C(S) of strictly convex real projective structures on S is a cell of dimension 2. We will describe a compactification of C(S) via Gromov-Hausdorff limits of the Hilbert metric. This work is joint with Daryl Cooper, Darren Long and MorwenThistlethwaite.

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Boundary metrics on convex cores of globally hyperbolic AdS 3 manifolds
Boubacar Diallo, University Paul Sabatier, France


We introduce global hyperbolicity in 3 dimensional AdS Geometry and recall Mess' parametrization of maximal, Cauchy compact, globally hyperbolic AdS 3 manifolds, analogous to Bers simultaneous uniformization theorem for quasifuchsian hyperbolic manifolds. We then give a partial answer to a conjecture related to the boundary metrics on convex cores of such manifolds.

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Lorentzian geometry
Todd Drumm, Howard University, USA


In these talks, we examine Lorentzian geometries of constant curvature; including Anti-deSitter (negative curvature) and de Sitter (postively curved) space. We will also introduce the conformal compactification of Minkowski space, the Einstein Universe. Additionally, we will look at causality issues associated to Lorentzian geometries.

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Projective structures and their holonomy representations
David Dumas, University of Illinois, Chicago, USA


We will define complex projective structures on surfaces and describe the moduli space that parametrizes them. We will then discuss the relationship between complex projective structures and their holonomy representations, as seen through properties of the holonomy map from the moduli space to the PSL(2,C) character variety.

The complicated structure of this map contrasts sharply with the situation for some other low-dimensional geometric structures, such as hyperbolic structures on compact manifolds, and a number of basic questions remain unanswered. However, we will describe some ways in which the theory of complex projective structures has been successfully applied to problems in Teichmuller theory and Kleinian

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Nielsen realization problem for asymptotic Teichmuller modular groups
Ege Fujikawa, Chiba University, Japan


We prove that every finite subgroup of the asymptotic Teichmuller modular group has a common fixed point in the asymptotic Teichmuller space under a certain geometric condition of a Riemann surface, and give the answer of an asymptotic version of the Nielsen realization problem.

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Higgs bundles and Hermitian symmetric spaces
Oscar Garcia-Prada, Consejo Superior de Investigaciones Científicas, Spain


We show how Higgs bundles on a compact Riemann surface can be used to study representations of the fundamental group of the surface into the isometry group of a non-compact Hermitian symmetric space.

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Measurable Riemann mapping theorem
Jonah Gaster, University of Illinois at Chicago, USA


A presentation of elementary material to understand the classical proof and utility of the so-called Measurable Riemann Mapping Theorem, due to Lars Ahlfors and Lipman Bers. If time permits, connections to Teichmuller theory will be explored.

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3-dimensional affine space forms and hyperbolic geometry
William Goldman, University of Maryland, USA


Flat Riemannian manifolds arise quotients of Euclidean space by discrete groups of isometries, and correspond to classical crystallographic groups. Such structures can equivalently be defined as systems of local coordinates into affine space where the coordinate changes are locally isometries. The theorems of Bieberbach provide an effective classification of such structures. Analogous questions for manifolds with flat connections, or equivalently, quotients by groups of affine transformations are considerably more difficult, and presently unsolved. In this talk I will describe how the classification in dimension three, reduces to a question on hyperbolic geometry on open 2-manifolds. This represents joint work with Fried, Drumm, Margulis, Labourie, Charette and Minsky.

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Generalized Teichmueller spaces and moduli spaces of geometric structures
Olivier Guichard, Université Paris-Sud, Orsay, France


In the space of representations of a surface group into a Lie group $G$, there are sometimes connected components that resemble to the classical Teichmueller space (defined for $G=SO(2,1)$): this resemblance is either topological or by an anologous definition. Those connected components are now usually called Generalized Teichmueller Spaces. Indeed recents results say that the representations in these spaces have many properties similar to the representations in the classical Teichmueller space. In this talk I will address the problem to describe the generalized Teichmueller spaces as moduli spaces of geometric structures. This is a joint work with Anna Wienhard.

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Harmonic deformations of hyperbolic structures
Craig Hodgson, The University of Melbourne, Australia


We survey deformation theory for hyperbolic manifolds with cone-type singularities, and some of its applications. Representing infinitesimal deformations by harmonic forms leads to rigidity and non-rigidity theorems, as well as effective estimates on the change in geometry as hyperbolic structures are deformed. Applications include the study of hyperbolic Dehn surgery on 3-manifolds, Kleinian groups, and the geometry of convex polyhedra in hyperbolic 3-space.

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Linear slices close to a Maskit slice
Kentaro Ito, Nagoya University, Japan


We consider linear slices of the deformation space of Kleinian once-punctured torus groups. We will show that when complex numbers a_n converge to 2 the corresponding linear slices converge to the Maskit slice if a_n --> 2 horocyclically
and to a proper subset of the Maskit slice if a_n --> 2 tangentially, where the proper subset is the intersection of translations of the Maskit slice. We will also mention the relation between this result and the complex Fenchel-Nielsen coordinate and the complex probability.

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Infinitely many Dehn fillings with compressing surfaces
Pradthana Jaipong, University of Illinois at Urbana-Champaign, USA


A closed totally geodesic surface in the figure eight knot complement remains incompressible in all but finitely many Dehn fillings. In this paper, we show that there is no universal upper bound on the number of such fillings, independent of the surface. This answers a question of Ying-Qing Wu.

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Primitive stable representations and Cannon-Thurston maps
Woojin Jeon, Seoul National University, Korea


I will talk about Minsky's conjecture about `primitive stable representations' and the use of Cannon-Thurston map for solving it. The talk would include some basic notions of hyperbolic 3-manifolds but I want to concentrate more on Cannon-Thurston maps.

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A tale of two spaces: moduli spaces of Riemann surfaces and locally symmetric spaces
Lizhen Ji, University of Michigan, USA


It is known that mapping class groups of surfaces and arithmetic subgroups of semisimple Lie groups share many common properties, and comparison between them has motivated many problems and produced a lot of results. In this talk, I will discuss some similarities and interaction between two classes of spaces related to them: moduli spaces of Riemann surfaces and arithmetic locally symmetric spaces.

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Transitional geometry
Steven Kerckhoff, Stanford University, USA


This will be a survey of results involving the process of moving between types of geometric structures. The most familiar of these is the interpolation between hyperbolic and spherical geometries, passing through Euclidean geometry. Other examples will also be described, including an explanation of a transition between hyperbolic and anti deSitter geometry in dimension 3.

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Surface group representation in semisimple Lie groups
Inkang Kim, Korea Institute for Advanced Study, Korea


We consider surface group representations in semisimple Lie groups to study their deformability and rigidity in terms of Toledo invariant and group cohomology initiated by Weil. We will describe a criterion for deformability to Zariski dense ones and as a corollary, we will prove that they are rigid only if the images lie in the groups associated to tube type Hermitian symmetric spaces. If time permits, we can discuss about some local rigidity phenomena of the other lattices as well.

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Pseudo-Anosovs with small dilatation and the Dehn fillings of the magic manifold
Eiko Kin, Tokyo Institute of Technology, Japan


We consider pseudo-Anosovs which occur as the monodromies on fibers for Dehn fillings of the so called magic manifold. By using these pseudo-Anosovs, we discuss the questions on the minimal dilatation for pseudo-Anosov homeomorphisms (with orientable invariant foliations). This is joint work with Mitsuhiko Takasawa.

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Lattices, mapping class groups and right-angled Artin groups
Thomas Michael Koberda, Harvard University, USA


I will introduce some of the machinery needed to show an analogy between certain lattices in rank one semisimple Lie groups and mapping class groups. The analogy is that after taking an irredundant generating set for a discrete subgroup of the Lie group or a mapping class group and raising the generators to sufficiently large powers, one obtains a right-angled Artin group.

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Entropy versus volume for surface automorphisms
Sadayoshi Kojima, Tokyo Institute of Technology, Japan


To each surface automorphism, we can assign its topological entropy and simplicial volume of its mapping torus. These are well behaved numerical invariants which measure its complexity. We plan to give a survey on comparison theorems for these invariants developed in the last decade.

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An algebra of observables for cross ratios
Francois Labourie, Université Paris-Sud, Orsay, France


We define a Poisson Algebra called the swapping algebra using the intersection of curves in the disk. We interpret a subalgebra of the fraction swapping algebra -- called the algebra of multifractions -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions onthe Hitchin component as well as on the space of SL(n;R)-opers with trivial holonomy. We finally relate our Poisson structure to the Drinfel'd-Sokolov structure and to the Atiyah-Bott-Goldman symplectic structure for classical Teichmuller spaces and Hitchin components.

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Topology and singularities of free group character varieties
Sean Lawton, University of Texas at Pan American, USA


Let X be the moduli of SL(n,C), SU(n), GL(n,C), or U(n) valued representations of a rank r free group. We compute the fundamental group of X and show that these four moduli otherwise have identical higher homotopy groups. We then classify the singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Lastly, we show that the moduli X are generally not topological manifolds, except for a few examples we explicitly describe.

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Characteristic submanifolds of hyperbolic 3-manifolds
Cyril Lecuire, University Toulouse III, France


We will explain an alternate proof of results of Jaco-Shalen and Johannson on homotopy equivalences and homeomorphisms of 3-manifolds with the extra assumption that the manifold under consideration is hyperbolisable. A characteristic submanifold of a compact 3-manifold is, roughly speaking, a submanifold that contains all the essential annuli and tori. It plays a fundamental role in the JSJ decomposition. Using topological properties of the limit set, we construct such a caracteristic submanifold for a 3-manifold equiped with a complete hyperbolic metric. Then using more topological properties of the limit set, we explain the possible obstructions that can prevent a homotopy equivalence to be homotopic to a homeomorphism.

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Hyperbolic structures on surfaces
Chris Leininger, University of Illinois, UC, USA


We will cover some of the basic aspects of the theory of Teichmuller spaces, from the point of view of hyperbolic geometry. Starting with elementary hyperbolic geometry, we will describe the Teichmuller space, construct the Fenchel-Nielsen coordinates, study the action of the mapping class group, and end with a discussion of Thurston's compactification. We will not assume any prior knowledge of the topic.

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Symplectic structures on character varieties
Brice Loustau, University Paul Sabatier, France


After very briefly reviewing basic facts about symplectic geometry, I will describe the general construction of Goldman in "The Symplectic Nature of Fundamental Groups of Surfaces (1984)", which endows any character variety with a natural symplectic structure. I will then investigate these in the cases of Teichmuller space and the space of complex projective structures on a surface. Viewing this symplectic structure from various angles will provide insight about the geometry of these deformation spaces, while displaying some useful techniques in the analytic description of such spaces.

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Solving Thurston's equation on triangulated 3-manifolds
Feng Luo, Rutgers University, USA


Given a closed triangulated 3-manifold, when does there exist a solution to the Thurston's gluing equation for hyperbolic metrics? We will discuss various approaches to resolve this question in the talk. Our recent work on volume and circle-valued angle structure provides a possible way to answer the question.
Part of the talk is based on the joint work with Tian Yang and Stephan Tillmann.

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Hyperbolicity of the genus two Hatcher-Thurston complex
Jiming Ma, Fudan University, China


For the genus $1$ surfaces $F_{1,n}$, we show that the Hatcher-Thurston complex $\mathscr{HT}(F_{1,n})$ is hyperbolic. For the genus $2$ closed surface, we show that the Hatcher-Thurston complex $\mathscr{HT}(F_{2})$ is strongly relatively hyperbolic with respect to a family of its subspaces $\{X_{\gamma}\}$, where $\gamma$ ranges over all separating curves of $F_{2}$, and each $X_{\gamma}$ is isometric to the product of two Farey graphs. This is a joint work with Youlin Li.

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The topology of deformation spaces of Hyperbolic 3-manifolds
Aaron D Magid, University of Maryland, USA


For any 3-manifold M, the deformation space $AH(M)$ is the space of all marked hyperbolic 3-manifolds homotopy equivalent to $M$. This deformation space naturally inherits a topology as a subset of the $PSL(2,\mathbb{C})$ character variety of $\pi_1(M)$. After reviewing some of the classical results that describe the topology of the interior of $AH(M)$, we will show that in many cases there are points in the boundary where $AH(M)$ fails to be locally connected. These cases include when $M$ is homotopy equivalent to a surface of genus at least 2. This is a generalization of Ken Bromberg's result that the space of Kleinian punctured torus groups is not locally connected.

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Counterexamples to a simple loop conjecture for PSL(2,C)
Jason Manning, University at Buffalo, USA


Minsky asked: Does every non-faithful homomorphism from a closed orientable surface group to PSL(2,C) kill some simple loop? We show that the answer is no by producing examples of non-faithful homomorphisms sending every simple loop either to a non-trivial parabolic or to a loxodromic with non-real trace. This is joint work with Daryl Cooper.

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Mappings of finite distortion and Teichmuller spaces
Gaven Martin, Massey University, New Zealand


Mappings of finite distortion are generalisations of quasiconformal mappings. They allow much greater flexibility in studying degenerating structures. Thus there is the possibility of having a mapping in ones hand on the boundary of Teichm¨¹ller spaces (of various dynamical systems) and the dream of an analytic proof of the ELC. Typically the defining equations for these mappings are degenerate elliptic (quasiconformal <=> uniformly elliptic) and there are very interesting analytic problems involved in their study. Here we discuss the extremal theory of mappings whose distortion lies in an L^p space (or nearby Zygmund space). There are interesting connections with the theory of harmonic mappings and particularly the Nitsche conjecture (now Theorem) and the Schoen conjecture. We may talk about interesting applications in nonlinear materials science too, if time allows.

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Hyperbolic cone-manifolds with prescribed holonomy
Daniel Virgil Mathews, Université de Nantes, France


We examine the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A geometric cone-manifold structure on a surface, with all interior cone angles being integer multiples of 2¦Ð, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a geometric cone-manifold structure. Our constructions involve the Euler class of a representation, the universal covering group of the orientation-preserving isometries of the hyperbolic plane, and the action of the outer automorphism group on the character variety.

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Surfaces with large cone angles
Greg McShane, University of Grenoble I, France


We discuss the topology and dynamics of the modular group for the moduli space
of surfaces a cone angle between pi and 2*pi.
The case of the torus has already been studied by Tan and Zhang.

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The Gardiner-Masur boundary of Teichmueller space
Hideki Miyachi, Osaka University, Japan


The Gardiner-Masur boundary (GM-boundary, for short) is the boundary of Teichmueller space which collects the asymptotic behaviors of extremal lengths of simple closed curves. Any point of the GM boundary is realized as the projective class of a non-negative function on the set of homotopy classes of simple closed curves, like as points of the Thurston boundary. It is known that the GM boundary contains the Thuston boundary. Furthermore, corresponding non-negative function to any point of the GM boundary extends as a continuous function on the space of measured foliations.

A point of the GM boundary is said to be uniquely ergodic if its zero set contains a uniquely ergodic measured foliation. In this talk, I will observe that any uniquely ergodic boundary point of the GM boundary is represented by the intersection number function, and hence, it is contained in the Thurston boundary.

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On discreteness of commensurators
Mahan Mj, RKM Vivekananda University, India


We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor.In particular for rank one or simple Lie groups, Zariski dense subgroups with non-empty domain of discontinuity have discrete commensurators. This generalizes a Theorem of Greenberg for Kleinian groups.

We then prove that for all finitely generated, Zariski dense, infinite covolume discrete subgroups of $\Isom ({\mathbb{H}}^3)$, commensurators are discrete. Together these prove discreteness of commensurators for all known examples of finitely generated, Zariski dense, infinite covolume discrete subgroups of $\Isom(X)$ for $X$ an irreducible symmetric space of non-compact type.

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SL(2,R)-flows on the Teichmueller space
Gabriele Mondello, University of Roma "La Sapienza", Italy


We construct a family of SL(2,R) local flows on the cotangent space of the Teichmueller space that specialize to the well-known action of SL(2,R) on the space of quadratic differentials.

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Penner's coordinate-system for a representation space on a punctured surface group and its applications
Toshihiro Nakanishi, Shimane University, Japan


The lambda length of an ideal arc on a hyperbolic surface with punctures can be complexified so that it gives a function on the faithful representation space of the surface group into SL(2,C).

Using this fact we extend R. C. Penner 's work on the Teichmuller space to the representation space. One of the interesting features is that the ideal Ptolemy equation, which is a hyperbolic-geometric result and plays an important role in representing the mapping class group by a group of rational transformations, is derived from basic trace identities. We give also several applications of Penner's coordinate-system.

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On the topology of H(2)
Duc Manh Nguyen, Max-Planck-Institut fur Mathematik, Germany


H(2) is the moduli space of pairs (M, \omega), where M is a Riemann surface of genus two, and \omega is a holomorphic 1-form on M having only one zero of order 2, H(2) can be identified with the space of translation surfaces of genus two having only one singularity with cone angle 6\pi. C^* acts naturally on H(2), and the quotient M(2)=H(2)/C^* is the set of pairs (M,W), where M is a Riemann surface of genus two, and W is a marked Weierstrass point of M.

Using decompositions of surfaces in H(2) into parallelograms, we single out a proper subgroup \Gamma of Sp(4,Z) generated by 3 elements, and show that M(2) is homeomorphic to the quotient J_2/\Gamma, where J_2 is the Jacobian locus of Riemann surfaces in the Sigel upper half space H_2. A direct consequence of this is that [Sp(4,Z):\Gamma]=6. The group \Gamma can be also interpreted as the image of the fundamental group of M(2) in Sp(4,Z).

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Moduli spaces of hyperbolic surfaces with cone angles
Paul Norbury, University of Melbourne, Australia


The moduli space of hyperbolic surfaces with cone angles behaves much like the usual moduli space when the cone angles are less than pi but is still defined for cone angles greater than pi. In this talk I will discuss the behaviour of the moduli space as a cone angle tends towards 2pi.

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Connected components of the representation space
Frederic Palesi, Institute Fourier, Grenoble, France


In this talk, I will construct the Euler class of a representation in PSL(2,R) from a simple point of view. Then I will try to explain the main ideas behind Goldman's result on connected components. Finally, if time permits, I will try to generalize this to non-orientable surface groups.

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On the automorphism group of the ideal triangulation graph
Athanase Papadopoulos, University Louis Pasteur Strasbourg, France


We show that the automorphism group of the ideal triangulation graph of a surface with punctures is the extended mapping class group, except for a finite set of surfaces. I will also survey some other properties of this triangulation graph. (Joint work with Mustafa Korkmaz).

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Characters in complex hyperbolic geometry
John Parker, University of Durham, UK


Complex hyperbolic space is a natural generalisation of the more familiar (real) hyperbolic space, and discrete groups of complex hyperbolic isometries are natural generalisations of Fuchsian and Kleinian groups. In these lectures I will give a brief introduction to complex hyperbolic space and its isometries and then I will discuss how spaces of discrete groups may be parametrised using trace coordinates. I will focus on complex hyperbolic representation spaces of surface groups and generalised triangle groups.

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Compactification of moduli spaces of completely reducible representations
Anne Parreau, University Grenoble 1, France


The good quotient space $\mathcal{X}$ of representations of a f.g. group into a reductive group $G$ can be geometrically described using the notion of complete reducibility on the associated symmetric space $X$. The Cartan projection can be seen as a distance on $X$ with values in a closed Weyl chamber $\mathfrak{C}$ of $X$. We describe the natural compactification of $\mathcal{X}$ based on the $\mathfrak{C}$-valued length spectra.

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Regenerating hyperbolic cone 3-manifolds from dimension 2
Joan Porti, Universitat Autònoma de Barcelona, Spain


Let P be a hyperbolic polygon. We view it as a 2-orbifold with mirror boundary. Let O be a Seifert fibered 3-orbifold, with space of leafs isomorphic to P. We prove that O admits a family of hyperbolic cone structures, with singular locus the ramification locus, and with cone angles less that pi. These structures are obtained by deforming the degenerate hyperbolic structure on P, provided that we choose the poligon of minimal perimeter among those with given angles.

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Surface groups, 3-manifold groups and SL_3
Alan Reid, University of Texas, Austin, USA


This talk will describe recent work on using representations of 3-manifold groups and surface groups to study the subgroup structure (both finite and infinite index) of SL(3,Z).

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A variation of McShane's identity for 2-bridge links
Makoto Sakuma, Hiroshima University, Japan


This is a joint work with Donghi Lee (Pusan University).
We prove a variation of McShane's identity for 2-bridge links, which describes the cusp shape of a hyperbolic 2-bridge link complement by an infinite series involving the complex lengths of certain family of closed geodesics.
The convergence of the infinite series is established as a consequence of the following results.
(1) Characterization of those essential simple loops on a 2-bridge sphere which are null-homotopic (resp. peripheral) in the link complement.
(2) A necessary and sufficient condition for two essential simple loops on a 2-bridge sphere to be homotopic in the link complement.
These results are proved by using the small cancellation theory.

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Some applications of 3-dimensional AdS geometry to Teichm\"uller theory
Jean-Marc Schlenker, University Toulouse III, France


We will expose several recent work (joint with Francesco Bonsante) where 3-dimensional AdS geometry is used to obtain new results in Teichm\"uller theory. One result is that the composition of the left earthquakes on two measured laminations that fill a surface has a fixed point in Teichm\"uller space. Another is that quasi-symmetric homeomorphism of the circle has a unique quasiconformal minimal Lagrangian extension to the hyperbolic disk. Finally we will describe an extension of the Earthquake flow.

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Kerckhoff's lines of minima in Teichmueller space
Caroline Series, University of Warwick, UK


In 1992, Kerckhoff introduced the idea of a line of minima in the Teichmueller space of a surface S, namely a line along which a convex combination of the lengths of a pair of geodesic laminations on S are minimised. This survey will introduce the basic facts about lines of minima and then discuss their connection with quasifuchsian groups which are almost Fuchsian and their relationship to Teichmueller geodesics.

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On holomorphic families of punctured tori
Hiroshige Shiga, Tokyo Institute of Technology, Japan


We will give an overview of a relationship between holomorphic families of Riemann surfaces and the theory of Teichmuller spaces, and show some results on holomorphic families of tori with punctures.

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Generators and symplectic structure of character varieties
Adam Sikora, University at Buffalo, USA


In the first part of the talk we will discuss finite generating sets of the coordinate rings of G-character varieties, for various algebraic groups G. This is a subtle question for G=SO(n) and for exceptional groups. Let M be a compact 3-manifold with connected boundary F. It is known that for every reductive G, X_G(F) is a symplectic manifold (with singularities). We will discuss the question whether the image of X_G(M) in X_G(F) is Lagrangian. There is a some confusion concerning this issue in the literature.

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The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold
Teruhiko Soma, Tokyo Metropolitan University, Japan


We show that the connected component of the identity map in the diffeomorphism group of a non-Haken Seifert fibered space with hyperbolic base orbifold is homotopy equivalent to the circle. By using this fact, we prove that the inclusion from the isometry group of the Seifert fibered space to the diffeomorphism group is a homotopy equivalence. This is a joint work with Darryl McCullough.

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Supersymmetric surface operators, four-manifold theory and invariants in various dimensions
Meng-Chwan Tan, National University of Singapore


I will discuss the application of supersymmetric surface operators in a topologically-twisted N=2 pure SU(2) gauge theory to the study of four-manifolds and topological invariants in two, three and four dimensions. I will explain how elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsvath-Szabo and Taubes, can be derived. I will also explain how one can obtain mathematically novel identities among the Gromov and singular Seiberg-Witten invariants of symplectic four-manifolds from the underlying physics, and elucidate how these invariants can be related to the instanton and monopole Floer homologies of certain three-submanifolds. Last but not least, I will present new and economical ways of deriving and understanding various important mathematical results concerning (1) knot homology groups from singular instantons by Kronheimer-Mrowka; (2) monopole Floer homology by Kutluhan-Taubes; and (3) Seiberg-Witten theory on symplectic four-manifolds.

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Graph manifolds have virtually positive Seifert volume
Shicheng Wang, Beijing University, China


The Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence for each closed orientable prime 3-manifold N, the set of mapping degrees D(M,N) is finite for any 3-manifold M unless N is finitely covered by either a torus bundle, or a trivial circle bundle, or the 3-sphere. This is joint work with Pierre Derbez.

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Character varieties and Morse theory
Richard Wentworth, University of Maryland, USA


For compact symplectic manifolds with a hamiltonian action by a connected, compact Lie group G, a theorem of Kirwan states that the Morse theory of the square of the moment map gives a stratification that is perfect for G-equivariant cohomology. Atiyah-Bott showed that the space of unitary connections on a bundle over a Riemann surface gives an infinite dimensional example of this phenomenon. In this talk I will discuss some recent results on certain singular versions of this picture obtained when one looks at bundles coupled with a holomorphic section of an associated bundle. Specific examples include Higgs bundles and Bradlow pairs, and these are related to character varieties of surface groups. I will show how calculations of equivariant cohomology of moduli spaces is still possible, even though the Morse stratification fails in general to be perfect.

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Polynomial pick forms for affine spheres
Mike Wolf, Rice University, USA


We discuss affine spheres modeled on the plane whose Pick forms are ploynomials. These arise in degenerating sequences of convex projective structures on surfaces. (Joint work with David Dumas.)

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Connected components of the compactification of representation spaces
Maxime Wolff, Université de Paris 6, France


The connected components of the representation space of a surface group in $PSL(2,\R)$ have been classified by W. Goldman. This representation space has been compactified by Morgan and Shalen, thus extending the Thurston compactification of Teichmuller spaces. The topological behaviour of the Thurston compactification of Teichmuller space is well-known, and in this talk, we will discuss the behaviour of this compactification at the boundary of the other connected components.

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Understanding Weil-Petersson curvature
Scott Wolpert, University of Maryland, USA


A report is presented on the program to describe the intrinsic geometry of the Weil-Petersson metric and geodesic-length functions. Formulas for the metric, covariant derivative and the curvature tensor are presented. A lower bound for sectional curvature in terms of the reciprocal systole and a classification of asymptotically flat multi sections are discussed. The method of estimating distant sums in hyperbolic geometry is sketched. Recent applications involving curvature are discussed, including results of Cavendish-Parlier, Liu-Sun-Yau, the Burns-Masur-Wilkinson ergodicity result and the Yamada model metric. A research report is available at arXiv:0809.3699 and an introduction is found in the author¡¯s AMS-CBMS-113 lectures Families of Riemann Surfaces and Weil-Petersson Geometry.

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Cohomologies, Abelian and non-Abelian
Eugene Xia, National Cheng Kung University, Taiwan


These lectures give a basic outline of the construction of the representation varieties, the moduli of flat connections and of Higgs bundles over projective varieties in characteristic zero. The emphasis is on Riemann surfaces and the symplectic structures on these moduli spaces with a view toward applications in dynamics of the mapping class group actions.

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Computer experiments on once punctured torus groups
Yasushi Yamashita, Nara Women's University, Japan


We consider a simple class of Kleinian groups called once punctured torus groups. In this tutorial, we will show how to make a computer program from scratch that can visualize fundamental sets and limit sets of the groups. No knowledge of computer programming is assumed.

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An invitation to elementary hyperbolic geometry
Ying Zhang, Suzhou University, China


This tutorial consists of four lectures. In Lecture 1 we review the contents of Book I of Euclid's Elements and explore the so-called neutral geometry. In Lecture 2 we study hyperbolic plane geometry in the spirit of the discoverers of non-Eulidean geometry, making no use of models of the hyperbolic plane. In Lecture 3 we classify the isometries of the hyperbolic plane and study the structure of the group of isometries. In Lecture 4 we study the hyperbolic trigonometry by considering identities of isometries associated to a triangle and working out the formulas in the upper half-plane model of the hyperbolic plane.

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Delambre-Gauss formulas in hyperbolic 4-space
Ying Zhang, Suzhou University, China


In this joint work with Ser Peow Tan and Yan Loi Wong, we study oriented, augmented, right-angled hexagons the hyperbolic 4-space H^4, via Clifford numbers or quaternions. We first define the quaternion half side-lengths whose angular parts are half the Euler angles associated to a certain orientation preserving isometry of the Euclidean 3-space. This generalizes the complex half side-lengths of oriented
right-angled hexagons in H^3. We further explain how to geometrically read off the quaternion side-lengths for a given oriented, augmented, right-angled hexagon in H^4. We obtain generalized Delambre-Gauss formulas for oriented right-angled hexagons in H^3 and, as the main result, generalized Delambre-Gauss formulas for oriented, augmented, right-angled hexagons in H^4, both of which generalize the classical Delambre-Gauss formulas for spherical and hyperbolic triangles.

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