Geometry, Topology and Dynamics of Character Varieties
(18 Jun - 15 Aug 2010)
## ~ Abstracts ~
Ilesanmi Adeboye, University of California, USAWe 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.
Javier Aramayona, National University of Ireland at Galway, IrelandWe 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.
Christopher Atkinson, Temple University, USAWe 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.
Yves Benoist, Université Paris-Sud, Orsay, FranceLet 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.
Ian Biringer, Yale University, USALet 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.
Francesco Bonsante, University of Pavia, ItalyLet 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.
Steve Boyer, Université du Québec à Montréal, CanadaIn 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.
Steven Bradlow, University of Illinois, UC, USAWe 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.
Jeff Brock, Brown University, USAThe 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.
Kenneth Bromberg, University of Utah, USAIn 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.
Richard Brown, Johns Hopkins University, USAFor 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.
Richard Canary, University of Michigan, USAIf 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.
Serge Cantat, University de Rennes I, FranceI 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.
Virginie Charette, Université de Sherbrooke, CanadaLet 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.
Suhyoung Choi, KAIST, KoreaWe 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.
Yves Cornulier, CNRS, University of Rennes, FranceIf 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.
Kelly Delp, Buffalo State College, USAA 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.
Boubacar Diallo, University Paul Sabatier, FranceWe 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.
Todd Drumm, Howard University, USAIn 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.
David Dumas, University of Illinois, Chicago, USAWe 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
Ege Fujikawa, Chiba University, JapanWe 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.
Oscar Garcia-Prada, Consejo Superior de Investigaciones Científicas, SpainWe 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.
Jonah Gaster, University of Illinois at Chicago, USAA 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.
William Goldman, University of Maryland, USAFlat 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.
Olivier Guichard, Université Paris-Sud, Orsay, FranceIn 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.
Craig Hodgson, The University of Melbourne, AustraliaWe 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.
Kentaro Ito, Nagoya University, JapanWe 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.
Pradthana Jaipong, University of Illinois at Urbana-Champaign, USAA 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.
Woojin Jeon, Seoul National University, KoreaI 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.
Lizhen Ji, University of Michigan, USAIt 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.
Steven Kerckhoff, Stanford University, USAThis 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.
Inkang Kim, Korea Institute for Advanced Study, KoreaWe 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.
Eiko Kin, Tokyo Institute of Technology, JapanWe 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.
Thomas Michael Koberda, Harvard University, USAI 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.
Sadayoshi Kojima, Tokyo Institute of Technology, JapanTo 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.
Francois Labourie, Université Paris-Sud, Orsay, FranceWe 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.
Sean Lawton, University of Texas at Pan American, USALet 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.
Cyril Lecuire, University Toulouse III, FranceWe 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.
Chris Leininger, University of Illinois, UC, USAWe 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.
Brice Loustau, University Paul Sabatier, FranceAfter 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.
Feng Luo, Rutgers University, USAGiven 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.
Jiming Ma, Fudan University, ChinaFor 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.
Aaron D Magid, University of Maryland, USAFor 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.
Jason Manning, University at Buffalo, USAMinsky 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.
Gaven Martin, Massey University, New ZealandMappings 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.
Daniel Virgil Mathews, Université de Nantes, FranceWe 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.
Greg McShane, University of Grenoble I, FranceWe 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.
Hideki Miyachi, Osaka University, JapanThe 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.
Mahan Mj, RKM Vivekananda University, IndiaWe 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.
Gabriele Mondello, University of Roma "La Sapienza", ItalyWe 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.
Toshihiro Nakanishi, Shimane University, JapanThe 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.
Duc Manh Nguyen, Max-Planck-Institut fur Mathematik, GermanyH(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).
Paul Norbury, University of Melbourne, AustraliaThe 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.
Frederic Palesi, Institute Fourier, Grenoble, FranceIn 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.
Athanase Papadopoulos, University Louis Pasteur Strasbourg, FranceWe 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).
John Parker, University of Durham, UKComplex 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.
Anne Parreau, University Grenoble 1, FranceThe 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.
Joan Porti, Universitat Autònoma de Barcelona, SpainLet 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.
Alan Reid, University of Texas, Austin, USAThis 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).
Makoto Sakuma, Hiroshima University, JapanThis 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.
Jean-Marc Schlenker, University Toulouse III, FranceWe 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.
Caroline Series, University of Warwick, UKIn 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.
Hiroshige Shiga, Tokyo Institute of Technology, JapanWe 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.
Adam Sikora, University at Buffalo, USAIn 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.
Teruhiko Soma, Tokyo Metropolitan University, JapanWe 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.
Meng-Chwan Tan, National University of SingaporeI 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.
Shicheng Wang, Beijing University, ChinaThe 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.
Richard Wentworth, University of Maryland, USAFor 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.
Mike Wolf, Rice University, USAWe 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.)
Maxime Wolff, Université de Paris 6, FranceThe 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.
Scott Wolpert, University of Maryland, USAA 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.
Eugene Xia, National Cheng Kung University, TaiwanThese 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.
Yasushi Yamashita, Nara Women's University, JapanWe 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.
Ying Zhang, Suzhou University, ChinaThis 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.
Ying Zhang, Suzhou University, ChinaIn 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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