Séminaire "Géométrie et groupes discrets"
Un lundi par mois à l'IHES. Les exposés sont entrecoupés d'une pause thé.
Contact : kassel A ihes.fr
Lundi 15 mai 2023, Amphithéâtre Léon Motchane
- 14h-15h15 : Simion Filip (University of Chicago)
- Titre : Anosov representations, Hodge theory, and Lyapunov exponents
- Résumé : Discrete subgroups of semisimple Lie groups arise in a variety of contexts, sometimes "in nature" as monodromy groups of families of algebraic manifolds, and other times in relation to geometric structures and associated dynamical systems. I will explain a method to establish that monodromy groups of certain variations of Hodge structure give Anosov representations, thus relating algebraic and dynamical situations. Among many consequences of these interactions, I will explain some uniformization results for domains of discontinuity of the associated discrete groups, Torelli theorems for certain families of Calabi-Yau manifolds (including the mirror quintic), and also a proof of a conjecture of Eskin, Kontsevich, Möller, and Zorich on Lyapunov exponents. The discrete groups of interest live inside the real linear symplectic group, and the domains of discontinuity are inside Lagrangian Grassmanians and other associated flag manifolds. The necessary context and background will be explained.
- 16h-17h15 : James Farre (Universität Heidelberg)
- Titre : Circle valued tight maps and horocycle orbit closures in Z-covers
- Résumé : The horocycle flow on hyperbolic surfaces has attracted considerable attention in the last century. In the '30s, Hedlund proved that all horocycle orbits are dense in closed hyperbolic surfaces, and the classification problem for horocycle orbit closures has been solved for geometrically finite surfaces. We are interested in the topology and dynamics of horocycle orbits in the geometrically infinite setting, where our understanding is much more limited.
- In this talk, I will discuss joint work with Or Landesberg and Yair Minsky: we give the first complete classification of orbit closures for a class of Z-covers of closed surfaces. Our analysis is rooted in a seemingly unrelated geometric optimization problem: finding a best Lipschitz map to the circle. We then relate the topology of horocycle orbit closures with the dynamics of the minimizing lamination of maximal stretch, as studied by Guéritaud-Kassel and Daskalopoulos-Uhlenbeck.
Lundi 3 avril 2023, Amphithéâtre Léon Motchane
- 14h-15h15 : Pierre-Emmanuel Caprace (Université Catholique de Louvain)
- Titre : New Kazhdan groups with infinitely many alternating quotients
- Résumé : Property (T) is a fundamental notion introduced by D. Kazhdan in the mid 1960's, that found numerous applications since then, notably in the context of rigidity of group actions. For a group G generated by a finite set S, property (T) means that there is a constant K>0 such that given any unitary representation of G on a Hilbert space without non-zero invariant vectors, every unit vector is displaced by some element of S to a point that is at least K apart. Finite groups have that property. Kazhdan proved that lattices in simple Lie groups of rank at least 2 all do as well.
- I will introduce a new class of infinite groups enjoying Kazhdan's property (T) and admitting alternating group quotients of arbitrarily large degree. Those groups are constructed as automorphism groups of the ring of polynomials in n indeterminates with coefficients in the finite field of order p, generated by a suitable finite set of polynomial transvections. As an application, we obtain explicit presentations of hyperbolic Kazhdan groups with infinitely many alternating group quotients, and explicit generating pairs of alternating groups of unbounded degree giving rise to expander Cayley graphs. The talk is based on joint work with Martin Kassabov.
- 16h-17h15 : Nicolas de Saxcé (CNRS & Université Paris-Nord)
- Titre : Rational approximations to linear subspaces
- Résumé : Dirichlet's theorem in Diophantine approximation implies that for any real x, there exists a rational p/q arbitrarily close to x such that |x-p/q| < 1/q2. In addition, the exponent 2 that appears in this inequality is optimal, as seen for example by taking x=. In 1967, Wolfgang Schmidt suggested a similar problem, where x is a real subspace of Rd of dimension ℓ, which one seeks to approximate by a rational subspace v. Our goal will be to obtain the optimal value of the exponent in the analogue of Dirichlet's theorem within this framework. The proof is based on a study of diagonal orbits in the space of lattices in Rd.
Lundi 6 mars 2023, Amphithéâtre Léon Motchane
- 14h-15h15 : Emmanuel Breuillard (University of Oxford)
- Titre : On the inverse problem for isometry groups of norms
- Résumé : We study the problem of determining when a compact group can be realized as the group of isometries of a norm on a finite-dimensional real vector space. This problem turns out to be difficult to solve in full generality, but we manage to understand the connected groups that arise as connected components of isometry groups. The classification we obtain is related to transitive actions on spheres (Borel, Montgomery-Samelson) on the one hand and to prehomogeneous spaces (Vinberg, Sato-Kimura) on the other. Joint work with Martin Liebeck, Assaf Naor and Aluna Rizzoli.
- 16h-17h15 : Jialun Li (CNRS & École Polytechnique)
- Titre : Exponential mixing of frame flows for geometrically finite hyperbolic manifolds
- Résumé : Let M be a geometrically finite hyperbolic manifold, that is, a hyperbolic manifold with a fundamental domain consisting of a finitely-sided polyhedron. There exists a unique measure on the unit tangent bundle invariant under the geodesic flow with maximal entropy, and we consider its lift to the frame bundle. In joint work with Pratyush Sarkar and Wenyu Pan, we prove that the frame flow is exponentially mixing with respect to this measure. To establish exponential mixing, we base ourselves on the countable coding of the flow and a version of Dolgopyat's method, à la Sarkar-Winter and Tsujii-Zhang. To overcome the difficulty of the fractal structure in applying Dolgopyat's method, we prove a large deviation property for symbolic recurrence to the large subsets.
Lundi 6 février 2023, Amphithéâtre Léon Motchane
- 14h-15h15 : Anja Randecker (Universität Heidelberg)
- Titre : Big mapping class groups as Polish groups
- Résumé : Classical mapping class groups, i.e. for surfaces of finite type, are well-studied but they are not particularly interesting from the point of view of topological groups as they are discrete.
- When we turn our attention to surfaces of infinite type, the situation changes drastically: in particular, the mapping class groups are now "big" (here: uncountable) and we can define an interesting (here: non-discrete) topology on them. In particular, big mapping class groups are Polish groups and we can ask many new questions such as on automatic continuity or their (large-scale) geometry.
- In this talk, I will give an introduction to surfaces of infinite type and big mapping class groups and then focus on the question of topological behaviour of conjugacy classes. The second part is based on joint work with Jesús Hernández Hernández, Michael Hrušák, Israel Morales, Manuel Sedano, and Ferrán Valdez, and will feature tools from model theory in the proofs.
- 16h-17h15 : Thibault Lefeuvre (CNRS & IMJ-PRG)
- Titre : On frame flow ergodicity
- Résumé : The frame flow over negatively-curved Riemannian manifolds is a historical example of a partially hyperbolic dynamical system. Excluding some obvious counterexamples such as Kähler manifolds, its ergodicity was conjectured by Brin in the 70s. While it has been known since Brin-Gromov (1980) that it is ergodic on odd-dimensional manifolds (and dimension not equal to 7), the even-dimensional case is still open. In this talk, I will explain recent progress towards this conjecture: I will show that in dimensions 4k+2 the frame flow is ergodic if the Riemannian manifold is 0.27 pinched (i.e., the sectional curvature is between -1 and -0.27), and in dimensions 4k if it is 0.55 pinched. This problem turns out to be surprisingly rich and at the interplay of different fields: (partially) hyperbolic dynamical systems, algebraic topology (classification of topological structures over spheres), Riemannian geometry and harmonic analysis (Pestov identity and microlocal analysis). Joint work with Mihajlo Cekić, Andrei Moroianu, Uwe Semmelmann.
Lundi 16 janvier 2023, Amphithéâtre Léon Motchane
- 14h-15h15 : Jeffrey Danciger (University of Texas at Austin)
- Titre : Eigenvalue asymmetry for convex real projective surfaces
- Résumé : A convex real projective surface is one obtained as the quotient of a properly convex open set in the projective plane by a discrete subgroup of SL(3,R), called the holonomy group, that preserves this convex set. The most basic examples are hyperbolic surfaces, for which the convex set is an ellipse, and the holonomy group is conjugate into SO(2,1). In this case, the eigenvalues of elements of the holonomy group are symmetric. More generally, the asymmetry of the eigenvalues of the holonomy group is a natural measure of how far a convex real projective surface is from being hyperbolic. We study the problem of determining which elements (and more generally geodesic currents) may have maximal eigenvalue asymmetry. We will present some limited initial results that we hope may be suggestive of a bigger picture. Joint work with Florian Stecker.
- 16h-17h15 : Pierre Py (CNRS & Université de Strasbourg)
- Titre : Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices
- Résumé : Following C.T.C. Wall, we say that a group G is of type Fn if it admits a classifying space which is a CW complex with finite n-skeleton. For n = 2, one recovers the notion of being finitely presented. We prove that in a cocompact complex hyperbolic arithmetic lattice with positive first Betti number, deep enough finite index subgroups admit plenty of homomorphisms to Z with kernel of type Fm-1 but not of type Fm. This provides many non-hyperbolic finitely presented subgroups of hyperbolic groups and answers an old question of Brady. This is based on a joint work with C. Llosa Isenrich.
Lundi 12 décembre 2022, Amphithéâtre Léon Motchane
- 14h-15h15 : Jean Raimbault (Aix-Marseille Université)
- Titre : Thin parts of arithmetic locally symmetric spaces
- Résumé : The well-known collar or Margulis lemma describes the structure of negatively curved manifolds at mesoscopic scale, in particular it allows to describe these manifolds globally through the "thick-thin decomposition". This is not sufficient, however, to completely understand the homotopy type of the manifold, even roughly. In this talk I will describe an "arithmetic Margulis lemma" (essentially a consequence of work of E. Breuillard) which allows to describe thin parts at a macroscopic scale in certain circumstances, and how to use it to obtain sharp bounds on the volume of thin parts of arithmetic locally symmetric spaces. This is joint work with M. Frączyk and S. Hurtado.
- 16h-17h15 : Greg Kuperberg (UC Davis & IHES)
- Titre : Effective approximation in densely generated Lie groups
- Résumé : If a finite set S densely generates a compact, semisimple Lie group G, then how well does the set of words of length ℓ in S (and S-1) approximate G? We could ask for them words to be an ε-net of G; or, beyond an ε-net, we could ask for the words to be evenly distributed down to a scale of ε; or we could ask for an efficient algorithm to produce a word that lies within ε of any given g in G. An optimal statistical result, with ℓ = O(log 1/ε), was first established by Lubotzky, Phillips, and Sarnak when G = SU(2) for special choices of S; and later generalized by others, but still with some restrictions on S. Not long afterwards, in the context of quantum computing, Solovay and Kitaev independently established an algorithm to find a word with ℓ = O((log 1/ε)a) for any S and (initially) also G = SU(2). I will discuss the current status of different versions of this question, including versions when G might not be compact or S-1 might not be used. I will also discuss my own result, in which I improve the exponent in the (algorithmic) Solovay-Kitaev theorem from the previous best value of a = 3+δ to a = (logφ 2) + 1 + δ < 2.4405.
Lundi 7 novembre 2022, Amphithéâtre Simons (attention, salle inhabituelle)
- 14h-15h15 : Sebastian Hensel (Universität München)
- Titre : Hyperbolics and parabolics in the fine curve graph
- Résumé : The fine curve graph is a Gromov hyperbolic graph on which the homeomorphism group of a surface acts. It allows to apply tools from geometric group theory and the theory of mapping class groups in this setting.
- In this talk, we will describe the first entries in a dictionary linking dynamical properties of homeomorphisms acting on the surface to the geometry of the action on the fine curve graph. Furthermore, we will discuss phenomena not encountered in the setting of "classical" curve graphs — namely, homeomorphisms acting as parabolic isometries. This is joint work with Jonathan Bowden, Katie Mann, Emmanuel Militon and Richard Webb.
- Time permitting, I will describe ongoing work with Jonathan Bowden and Richard Webb concerning the Gromov boundary of the fine curve graph.
- 16h-17h15 : Yilin Wang (IHES)
- Titre : Holography of the Loewner energy
- Résumé : The link between the hyperbolic geometry of 3-manifolds and the conformal metrics on their boundary has been explored extensively in the context of hyperbolic geometry and is also motivated by the AdS3/CFT2 correspondence. An elementary observation is that the group of Möbius transformations on the Riemann sphere coincides with the isometries of the hyperbolic 3-space H3. The Loewner energy is a Möbius-invariant quantity that measures the roundness of Jordan curves. It arises from large deviations of SLE0+ and is a Kähler potential on the universal Teichmüller space endowed with the Weil-Petersson metric. We show that the Loewner energy of a Jordan curve in the Riemann sphere equals the renormalized volume of a submanifold of H3 constructed using the Epstein surfaces associated with the hyperbolic metric on both sides of the curve. This is work in progress with Martin Bridgeman (Boston College), Ken Bromberg (Utah), and Franco Vargas-Pallete (Yale).
Lundi 3 octobre 2022, Amphithéâtre Léon Motchane
- 14h-15h15 : Serge Cantat (CNRS & Université de Rennes 1)
- Titre : Elementary properties of groups of polynomial automorphisms
- Résumé : Let G be a finitely generated group acting faithfully by linear transformations on a finite-dimensional complex vector space.
The theorems of Malcev, Selberg, or Tits provide important properties satisfied by G.
To what extent do these properties continue to hold when G is acting by polynomial (instead of linear) transformations?
In order to address this question, I shall describe a few results that illustrate how one can use p-adic or finite fields for problems which are initially phrased in terms of complex numbers.
- 16h-17h15 : Çağrı Sert (Universität Zürich)
- Titre : Stationary measures for SL(2,R)-actions on homogeneous bundles over flag varieties
- Résumé : Let Xk,d denote the space of rank-k lattices in Rd. Topological and statistical properties of the dynamics of discrete subgroups of G=SL(d,R) on Xd,d were described in the seminal works of Benoist-Quint. A key step/result in this study is the classification of stationary measures on Xd,d. Later, Sargent-Shapira initiated the study of dynamics on the spaces Xk,d. When k ≠ d, the space Xk,d is of a different nature and a clear description of dynamics on these spaces is far from being established. Given a probability measure μ which is Zariski-dense in a copy of SL(2,R) in G, we give a classification of μ-stationary measures on Xk,d and prove corresponding equidistribution results. In contrast to the results of Benoist-Quint, the type of stationary measures that μ admits depends strongly on the position of SL(2,R) relative to parabolic subgroups of G. I will start by reviewing preceding major works and ideas. The talk will be accessible to a broad audience. Joint work with Alexander Gorodnik and Jialun Li.
Lundi 16 mai 2022, Amphithéâtre Léon Motchane
- 14h-15h15 : Kathryn Mann (Cornell University)
- Titre : Boundary rigidity for hyperbolic groups
- Résumé : A hyperbolic group acts on its Gromov boundary by homeomorphisms. In recent work with Jason Manning, we showed that for groups with sphere boundary, the boundary action is rigid in the sense of topological dynamics: any sufficiently small perturbation is semi-conjugate to the original action. In ongoing work also with Teddy Weisman, we are extending this result to all hyperbolic groups, using a coding argument in the spirit of Sullivan. This talk will introduce the rigidity problem and describe some of the tools towards the proof.
- 16h-17h15 : Adrien Boulanger (CNRS & Aix-Marseille Université)
- Titre : Large deviations of the escape rate of random walks on hyperbolic spaces
- Résumé : A group endowed with a probability measure comes naturally with a random walk. If moreover this group acts on some space, then one can push the random walk to the space under consideration, up to the choice of a basepoint. The resulting random walk is called the image random walk.
- In this talk, motivated by numerous examples (hyperbolic groups acting on one of their Cayley graphs, mapping class groups acting on the corresponding curve complex...), the (discrete) groups will act on Gromov hyperbolic spaces. We will discuss the escape rate for such image random walks, and more precisely the associated large deviations problem.
Lundi 11 avril 2022, Amphithéâtre Léon Motchane
- 14h-15h15 : Françoise Dal'Bo (Université Rennes 1)
- Titre : Foliations by hyperbolic surfaces and dynamics of the horocyclic flow
- Résumé : We will discuss topological and ergodic aspects of the horocyclic flow on families of examples. This is joint work with Fernando Alcalde Cuesta.
- 16h-17h15 : Bram Petri (IMJ-PRG)
- Titre : Extremal hyperbolic surfaces and the Selberg trace formula
- Résumé : The Selberg trace formula provides a link between the length spectrum and the Laplacian spectrum of a hyperbolic surface. I will speak about a joint project with Maxime Fortier Bourque in which we are using this formula to probe extremal problems in hyperbolic geometry. These are questions of the form: what is the hyperbolic surface of a given genus with the largest kissing number or the largest spectral gap? Concretely, I will explain the general principle of our method, which is inspired by ideas from the world of Euclidean sphere packings. Moreover, I will explain why the Klein quartic, the most symmetric Riemann surface in genus 3, solves one of our extremal problems.
Lundi 14 mars 2022, Amphithéâtre Léon Motchane
- 14h-15h15 : Colin Guillarmou (CNRS & Université Paris-Saclay)
- Titre : A mathematical approach to Liouville conformal field theory on Riemann surfaces
- Résumé : Conformal field theory is a vast subject intensively studied in theoretical physics since the 80s. In this talk I will explain how one can use probabilistic methods, analytic methods and tools from Teichmüller spaces and the geometry of Riemann surfaces to construct rigorously (in the mathematical sense) an important conformal field theory in dimension 2, called the Liouville conformal field theory. This theory is a theory of random Riemannian metrics on surfaces and its correlation functions can be computed explicitly and decomposed into two quantities: the so-called structure constant (the 3 point function on the sphere) and the Virasoro conformal blocks. The conformal blocks are holomorphic functions of the moduli of surfaces linked to the representation theory of the Virasoro algebra.
- This is based on joint works with Kupiainen, Rhodes and Vargas, and an ongoing work with the same authors together with Baverez.
- 16h-17h15 : Thomas Haettel (Université de Montpellier)
- Titre : Groups actions on injective metric spaces
- Résumé : A metric space is called injective if any family of pairwise intersecting balls has a non-empty intersection. Injective metric spaces enjoy many properties typical of nonpositive curvature. In particular, when a group acts by isometries on such a space, we will review the many consequences this has. We will also present numerous groups admitting an interesting action on an injective metric space, such as hyperbolic groups, cubulable groups, lattices in Lie groups, mapping class groups, some Artin groups...
Lundi 14 février 2022, Amphithéâtre Léon Motchane
- 14h-15h15 : François Labourie (Université Côte d'Azur)
- Titre : Positivity and representations of surface groups
- Résumé : Positivity is meant as a generalisation of the cyclic order on the circle. Associated to that is the notion of monotone maps from a cyclically ordered set in the circle.
- Generalisations of the idea of positivity appeared to be crucial in understanding some connected components of the space of representations of a surface group in a Lie group G, although the common phenomenon was not figured out until recently.
- In this talk, based on a preprint with Olivier Guichard and Anna Wienhard, I will start by examples generalizing this notion of cyclic order on the circle: convex curves or configurations in the plane, time like curve in Minkowski space. Then I will move to the general geometry of parabolic spaces and explain why the notion of positivity relates to special configurations of pairwise transverse triples and quadruples of points. This notion of positivity, which abides simple combinatorial properties, allows to define positive — or monotone — curves, then positive representations of surface groups.
- I will then sketch the proof of our main result: positive representations are Anosov and fill up connected components of the space of representations.
- 16h-17h15 : Bertrand Deroin (CNRS & Université de Cergy)
- Titre : Toledo Invariants of quantum representations
- Résumé : Quantum representations form a family of representations of modular groups of surfaces with values in projective pseudo-unitary groups PU(p,q), sending Dehn twists to finite-order elements. Toledo invariants of these representations, and more general characteristic classes, extend to cohomology classes defined on the Deligne-Mumford compactification of moduli spaces, defining cohomological field theories (CohFT). We will give explicit formulae for the Toledo part of the latter in some cases, including Fibonacci quantum representations. This allows to construct/recover complex hyperbolic structures on some moduli spaces. Work in progress with Julien Marché.
Lundi 17 janvier 2022, Amphithéâtre Léon Motchane
- 14h-15h15 : Sébastien Gouëzel (CNRS & Université Rennes 1)
- Titre : Ruelle resonances for geodesic flows on noncompact manifolds
- Résumé : Ruelle resonances are complex numbers associated to a dynamical system that describe the precise asymptotics of the correlations for large times. It is well known that this notion makes sense for smooth uniformly hyperbolic dynamics on compact manifolds. In this talk, I will consider the case of the geodesic flow on some noncompact manifolds. In a class of such manifolds (called SPR), I will explain that one can define Ruelle resonances in a half-plane delimited by a critical exponent at infinity. Joint work with Barbara Schapira and Samuel Tapie.
- 16h-17h15 : Maxime Wolff (IMJ-PRG)
- Titre : Automorphisms of fine graphs and some rotation properties
- Résumé : Recently Bowden, Hensel and Webb defined the fine graph of surfaces, extending the notion of curve graphs of surfaces to the study of homeomorphism or diffeomorphism groups of surfaces. Later Long, Margalit, Pham, Verbene and Yau proved that for a closed surface of genus g≥2, the automorphism group of the fine graph is no larger than the homeomorphism group of the surface, while Bowden, Hensel, Mann, Militon and Webb related properties of the rotation sets of torus homeomorphisms to their action on this fine graph. In undergoing joint work with Frédéric Le Roux, we extend the result of automorphism groups to the torus case, and in fact, to any surface of positive genus, and explore further the properties of homeomorphisms related to their action on the fine graph.
Lundi 13 décembre 2021, Amphithéâtre Simons (attention, salle inhabituelle)
- 14h-15h15 : Antonin Guilloux (IMJ-PRG & Institut Fourier)
- Titre : Limit set deformations inside the sphere at infinity of the complex hyperbolic plane
- Résumé : Surface group representations inside PU(2,1) share some geometric properties with Kleinian groups but also with real projective representations, related to divisible convex sets.
- A central object for understanding their properties is the limit sets in the sphere at infinity. I will present computer generated figures of these limit sets that help build some intuition for deformations of real Fuchsian representations. This leads to a new tool for working with these deformations, which we call thinness.
- Joint work (in progress) with E. Falbel and P. Will.
- 16h-17h15 : Rémi Coulon (CNRS & Université Rennes 1)
- Titre : Equations in periodic groups
- Résumé : The free Burnside group B(r,n) is the quotient of the free group of rank r by the normal subgroup generated by the n-th power of all its elements. It was introduced in 1902 by Burnside, who asked whether B(r,n) is necessarily a finite group or not. In 1968 Novikov and Adian proved that if r > 1 and n is a sufficiently large odd exponent, then B(r,n) is actually infinite. It turns out that B(r,n) has a very rich structure. In this talk we are interested in understanding equations in B(r,n). In particular we want to investigate the following problem: given a set of equations S, under which conditions does every solution to S in B(r,n) already come from a solution in the free group of rank r?
- Along the way we will explore other aspects of certain periodic groups (i.e. quotients of a free Burnside group) such as the Hopf / co-Hopf property, the isomorphism problem, their automorphism groups, etc.
- Joint work with Z. Sela.
Lundi 8 novembre 2021, Amphithéâtre Léon Motchane
- 14h-15h15 : Oscar García-Prada (ICMAT, Madrid)
- Titre : Higgs bundles and higher Teichmüller spaces
- Résumé : In this talk I will present a general construction in terms of Higgs bundles of the higher Teichmüller components of the character variety of a surface group for a real Lie group admitting a positive structure in the sense of Guichard-Wienhard. Key ingredients in this construction are the notion of magical sl(2)-triple, that we introduce, and the Cayley correspondence. Basics on Higgs bundle theory will be explained. (Based on joint work with Bradlow, Collier, Gothen and Oliveira.)
- 16h-17h15 : Federica Fanoni (CNRS & LAMA, Créteil)
- Titre : Isospectral hyperbolic surfaces of infinite genus
- Résumé : Two hyperbolic surfaces are said to be (length) isospectral if they have the same collection of lengths of primitive closed geodesics, counted with multiplicity (i.e. if they have the same length spectrum). For closed surfaces, there is an upper bound on the size of isospectral hyperbolic structures depending only on the topology. We will show that the situation is very different for infinite-type surfaces, by constructing large families of isospectral hyperbolic structures on surfaces of infinite genus.
Lundi 11 octobre 2021, Amphithéâtre Léon Motchane
- 14h-15h15 : Maria Beatrice Pozzetti (Universität Heidelberg)
- Titre : On Θ-positive surface subgroups in PO(p,q)
- Résumé : Surprisingly, there exist connected components of character varieties of fundamental groups of surfaces in semisimple Lie groups consisting only of injective representations with discrete image. Guichard and Wienhard introduced the notion of Θ-positive representations as a conjectural framework to explain this phenomenon. I will discuss joint work with Jonas Beyrer in which we establish several geometric properties of Θ-positive representations in PO(p,q). As an application, we deduce that they indeed form connected components of character varieties.
- 16h-17h15 : Uri Bader (Weizmann Institute of Science)
- Titre : Group Random Element Generators
- Résumé : In this talk I will discuss the notion of a GREG — a Group Random Element Generator — which is a generalization of a random walk on a group. Roughly, a Greg is a random sequence of group elements.
- Associated with a Greg one obtains a pair of Furstenberg-Poisson boundaries, the spaces of ideal futures and ideal pasts. An important property that a Greg might have is the Asymptotic Past And Future Independence. Gregs satisfying this property, namely Apafic Gregs, are very well behaved. Geodesic Flows in a negatively curved environment, as well as classical random walks on groups, give rise to Apafic Gregs. After surveying the subject, I will focus on linear representations of Gregs and the associated invariant called the Lyapunov spectrum. As it turns out, under mild assumptions the Lyapunov spectrum would be simple and continuously varying.
- The talk is based on joint work with Alex Furman.
Lundi 9 novembre 2020, Amphithéâtre Léon Motchane
Orateurs prévus : Bram Petri (IMJ-PRG) et Rémi Coulon (CNRS & Université de Rennes 1)
Séance annulée en raison du confinement
Lundi 12 octobre 2020, Amphithéâtre Léon Motchane
- 14h-15h15 : Andrés Sambarino (CNRS & IMJ-PRG)
- Titre : Discrete subgroups with Lipschitz limit set
- Résumé : In this talk we will focus on discrete subgroups Γ of higher rank Lie groups G whose limit set is a Lipschitz manifold, i.e. locally the graph of a Lipschitz map. This is a not-uncommon feature, verified by Zariski-dense groups, usually caused by stable geometric properties of the embedding Γ → G. We will recall several examples of such groups, specially those coming from Higher rank Teichmüller Theory. The main purpose of the lecture is to explain a recent result, in collaboration with B. Pozzetti and A. Wienhard, where we prove that the critical exponent of a specific combination of eigenvalues (only depending on the dimension of the limit set) is independent of the representation. We will also explore some of its consequences.
- 15h45-17h : Pascal Hubert (Aix-Marseille Université & CIRM)
- Titre : Some remarks on Novikov's problem for foliations on surfaces and Arnoux-Rauzy interval exchange transformations
- Résumé : I will discuss some questions related to Novikov's problem for foliations on surfaces.
- The problem is the following. Let M be a 3-periodic surface and H a plane intersecting M.
Which kind of curves are realized as the intersection of M and H?
- This problem was formulated by Sergey Petrovitch Novikov in the 80's. He conjectured that
the "trivial" situations (periodic and integrable) are generic.
I will discuss the simplest situation when M is very symmetric. In this case, with Dynnikov and Skripchenko,
we prove that the first return
of the foliation induced by H is an Arnoux-Rauzy interval exchange transformation.
I will give some properties of these maps (results with Arnoux, Cassaigne and Ferenczi).
In the most interesting situation, I will mention a work in progress with
Dynnikov, Mercat, Paris-Romaskevich and Skripchenko which is supposed to solve Novikov's conjecture.
- The talk will be accessible to a broad audience.
Lundi 30 mars 2020, Amphithéâtre Léon Motchane
Orateurs prévus : Andrés Sambarino (CNRS & IMJ-PRG) et Rémi Coulon (CNRS & Université de Rennes 1)
Séance annulée en raison du confinement
Lundi 2 mars 2020, Amphithéâtre Léon Motchane
- 14h30-15h45 : Jean-Philippe Burelle (Université de Sherbrooke)
- Titre : Local rigidity of diagonally embedded triangle groups
- Résumé : Recent work of Alessandrini-Lee-Schaffhauser generalized the theory of higher Teichmüller spaces to the setting of orbifold surfaces. In particular, these authors proved that, as in the torsion-free surface case, there is a "Hitchin component" of representations into PGL(n,R) which is homeomorphic to a ball. They explicitly compute the dimension of Hitchin components for triangle groups, and find that this dimension is positive except for a finite number of low-dimensional examples where the representations are rigid. In contrast with these results and with the torsion-free surface group case, we show that the composition of the geometric representation of a hyperbolic triangle group with a diagonal embedding into PGL(2n,R) or PSp(2n,R) is always locally rigid.
- 16h30-17h45 : Adrien Boyer (IMJ-PRG)
- Titre : Spherical functions on hyperbolic groups and property RD (Rapid Decay)
- Résumé : We investigate properties of some spherical fonctions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson-Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called property RD (Rapid Decay, also called Haagerup's inequality) as a particular case of a more general behavior of spherical functions on hyperbolic groups. Then I will explain how these representations are related to the so-called "complementary series". The problem of the unitarization of such representations will be at the heart of the discussion.
- If time permits, I will try to explain the idea of a constructive proof, using a boundary unitary representation, of a result due to de la Harpe and Jolissaint asserting that hyperbolic groups satisfy property RD.
Lundi 3 février 2020, Amphithéâtre Léon Motchane
- 14h30-15h45 : Peter Haissinsky (Aix-Marseille Université)
- Titre : Characterizations of Kleinian groups
- In low dimension, it is expected that topological properties determine a natural geometry. In this spirit, several characterizations are conjectured for Kleinian groups, i.e. discrete subgroups of PSL(2,C). We will survey different methods that lead to their topological and dynamical characterizations, and point out their limits and the difficulties encountered in obtaining a complete answer.
- 16h30-17h45 : Timothée Marquis (Université Catholique de Louvain)
- Titre : Cyclically reduced elements in Coxeter groups
- Résumé : Let W be a Coxeter group. We provide a precise description of the conjugacy classes in W, yielding an analogue of Matsumoto's theorem for the conjugacy problem in arbitrary Coxeter groups. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of W, thereby proving a conjecture of A. Cohen from 1994.
Lundi 6 janvier 2020, Amphithéâtre Léon Motchane
- 14h30-15h45 : Gérard Besson (CNRS & Université Grenoble Alpes)
- Titre : A Bishop-Gromov type inequality for some metric spaces
- The work concerns a δ-hyperbolic metric space (X,d) (possibly with extra conditions) and the main assumption is that its entropy, denoted by H, is bounded above. Then, if a subgroup of its isometry group acts properly and co-compactly and if D denotes the diameter of the quotient, we will show a Bishop-Gromov type inequality on (X,d) only in terms of δ, H and D. It is a curvature-free inequality and we will explain how the bound on the entropy plays the role of a (weak version of a) lower bound on the Ricci curvature and how the δ-hyperbolicity relates to a bound on the negative part of the sectional curvature. Some consequences of this inequality are a finiteness theorem as well as a compactness result.
- This is joint work with G. Courtois, S. Gallot and A. Sambusetti.
- 16h30-17h45 : Viktoria Heu (Université de Strasbourg)
- Titre : Algebraic isomonodromic deformations and the mapping class group
- Résumé : The mapping class group acts on the set of representations modulo conjugation of the fundamental group of n-punctured genus g-curves. For representations with values in SL(2,C), finite orbits of this action have been classified in the literature under various additional constraints. We complete this classification by the remaining case of reducible representations for g>0. This study is motivated by the following result: up to some minor technical conditions, representations modulo conjugation with values in GL(r,C) that have finite orbit under the action of the mapping class group are precisely those that appear as the monodromy of a logarithmic connection on the curve, with poles at the punctures, that admits an algebraic universal isomonodromic deformation. Both results concern a joint work with G. Cousin.
Lundi 9 décembre 2019, Amphithéâtre Léon Motchane
Orateurs prévus : Timothée Marquis (Université Catholique de Louvain) et Adrien Boyer (IMJ-PRG)
Séance annulée en raison du mouvement de grève
Lundi 18 novembre 2019, Amphithéâtre Léon Motchane
- 14h30-15h45 : Elisha Falbel (IMJ-PRG)
- Titre : Geometric structures modelled on closed orbits in flag manifolds
- Résumé : Path geometry and CR structures on real 3-manifolds were studied by E. Cartan. There is an interesting local geometry with curvature invariants and an interesting global geometry. In particular, one can obtain flat structures studying configurations of flags. The model spaces are closed orbits of SL(3,R) and SU(2,1) in a complex flag manifold. We will review these geometries and discuss a notion of flag structure, which includes both geometries. We also review volume and Chern-Simons invariants for such geometric structures.
- 16h30-17h45 : Harrison Bray (University of Michigan)
- Titre : Volume entropy rigidity in Hilbert geometries
- Résumé : In this talk we will discuss the Besson-Courtois-Gallot (BCG) theorem in the context of convex projective geometry. The BCG theorem is a rigidity statement relating the volume and entropy of a negatively curved Riemannian manifold, and has many applications including Mostow rigidity. In the world of convex real projective structures, the natural Hilbert geometry on these objects is only Finsler and the geometry is generally not even C2. We discuss our analogous BCG theorem and some applications in the case where the manifold is closed. We will include some ongoing work to extend the result to finite volume. This is based on joint work with Ilesanmi Adeboye and David Constantine.
Lundi 7 octobre 2019, Amphithéâtre Léon Motchane
- 14h30-15h45 : Romain Dujardin (LPMA, Sorbonne Université)
- Titre : Degenerations of SL(2,C) representations and Lyapunov exponents
- Résumé : Let G be a finitely generated group endowed with some probability measure μ and (ρλ) be a non-compact algebraic family of representations of G into SL(2,C). This gives rise to a random product of matrices depending on the parameter λ, so the upper Lyapunov exponent defines a function on the parameter space. Using techniques from non-Archimedean analysis and algebraic geometry, we study the asymptotics of the Lyapunov exponent when
λ goes to infinity. This is joint work with Charles Favre.
- 16h30-17h45 : Jérémy Toulisse (Université de Nice-Sophia Antipolis)
- Titre : Quasi-circles and maximal surfaces in the pseudo-hyperbolic space
- Résumé : Quasi-circles in the complex plane are fundamental objects in complex analysis; they were used by Bers to define an infinite-dimensional analogue of the usual Teichmüller space. After introducing the notion of quasi-circles in the boundary of the pseudo-hyperbolic space H2,n, I will explain how to construct a unique complete maximal surface in H2,n bounded by a given quasi-circle. This construction relies on Gromov's theory of pseudo-holomorphic curves and provides a generalization of maximal representations of surface groups into rank 2 Lie groups. This joint work with François Labourie and Mike Wolf.
Mardi 11 juin 2019, Amphithéâtre Léon Motchane
- 14h30-15h45 : Anton Zorich (IMJ-PRG)
- Titre : Bridges between flat and hyperbolic enumerative geometry
- Résumé : I will give a formula for the Masur-Veech volume of the moduli space of quadratic differentials in terms of psi-classes (in the spirit of Mirzakhani's formula for the Weil-Peterson volume of the moduli space of hyperbolic surfaces). I will also show that Mirzakhani's frequencies of simple closed hyperbolic geodesics of different combinatorial types coincide with the frequencies of the corresponding square-tiled surfaces. I will conclude with a (mostly conjectural) description of the geometry of a "random" square-tiled surface of large genus and of a "random" multicurve on a topological surface of large genus.
- The talk is based on joint work in progress with V. Delecroix, E. Goujard and P. Zograf. It is aimed at a broad audience, so I will try to include all necessary background.
- 16h30-17h45 : Ludovic Marquis (Université de Rennes 1)
- Titre : Geometrization of certain 4-dimensional groups
- Résumé : We consider discrete groups admitting proper cocompact topological actions by homeomorphisms on R4. We will say that such a group Γ is geometrized if we can build an action of Γ by projective transformations on a properly convex open subset of the real projective 4-space, or a convex cocompact action of Γ on the real hyperbolic 5-space or on its Lorentzian counterpart, the anti-de Sitter 5-space.
- Certain uniform lattices of the isometry group of hyperbolic 4-space are geometrizable by the three geometries mentioned above. We will discuss the existence of groups which are not uniform lattices in hyperbolic 4-space, and which yet admit several of these three geometries. If time allows, we will also discuss the corresponding deformation spaces.
- This is joint work with Gye-Seon Lee (Heidelberg).
Lundi 6 mai 2019, Amphithéâtre Léon Motchane
- 14h30-15h45 : Ursula Hamenstädt (Universität Bonn)
- Titre : Boundary actions and actions on Lp-spaces
- Résumé : We discuss some quite general properties of infinite discrete groups G acting on compact spaces. The spaces we mainly have in mind are horoboundaries of metric spaces which admit an isometric action of G. As an application, we show that the mapping class group of a surface of finite type admits a proper action on some Lp-space.
- 16h30-17h45 : Florent Schaffhauser (Universidad de los Andes & Université de Strasbourg)
- Titre : Higher Teichmüller spaces for orbifolds
- Résumé : The Teichmüller space of a compact 2-orbifold X can be defined as the space of faithful and discrete representations of the fundamental group π1(X) of X into PGL(2,R). It is a contractible space. For closed orientable surfaces, "higher analogues" of the Teichmüller space are, by definition, (unions of) connected components of representation varieties of π1(X) that consist entirely of discrete and faithful representations. There are two known families of such spaces, namely Hitchin representations and maximal representations, and conjectures on how to find others. In joint work with Daniele Alessandrini and Gye-Seon Lee, we show that the natural generalisation of Hitchin components to the orbifold case yields new examples of higher Teichmüller spaces: Hitchin representations of orbifold fundamental groups are discrete and faithful, and share many other properties of Hitchin representations of surface groups. However, we also uncover new phenomena, which are specific to the orbifold case.
Lundi 15 avril 2019, Amphithéâtre Léon Motchane
- 14h30-15h45 : Andrea Seppi (CNRS & Université Grenoble Alpes)
- Titre : Isometric embeddings of the hyperbolic plane into Minkowski space
- Résumé : Minkowski space of dimension 2+1 is the Lorentzian analogue of Euclidean 3-space. It is well-known that there exists an isometric embedding of the hyperbolic plane in Minkowski space, which is the analogue of the embedding of the round sphere in Euclidean space. However, differently from the Euclidean case, the embedding of the hyperbolic plane is not unique up to global isometries. In this talk I will discuss several results on the classification of these embeddings, and explain how this problem is related to Monge-Ampère equations, harmonic maps, and Teichmüller theory. This is joint work with Francesco Bonsante and Peter Smillie.
- 16h30-17h45 : Qiongling Li (Chern Institute of Mathematics, Nankai University)
- Titre : Harmonic maps for Hitchin representations
- Résumé : Hitchin representations are an important class of representations of fundamental groups of closed hyperbolic surfaces into PSL(n,R), at the heart of higher Teichmüller theory. Given such a representation j, there is a unique j-equivariant harmonic map from the universal cover of the hyperbolic surface to the symmetric space of PSL(n,R). We show that its energy density is at least 1 and that rigidity holds. In particular, we show that given a Hitchin representation, every equivariant minimal immersion from the hyperbolic plane into the symmetric space of PSL(n,R) is distance-increasing. Moreover, equality holds at one point if and only if it holds everywhere and the given Hitchin representation j is an n-Fuchsian representation.
Lundi 11 mars 2019, Amphithéâtre Léon Motchane
- 14h30-15h45 : Barbara Schapira (Université de Rennes 1)
- Titre : Dynamics of unipotent frame flows for hyperbolic manifolds
- Résumé : In joint work with François Maucourant, we study the dynamics of unipotent flows on frame bundles of hyperbolic manifolds of infinite volume. We prove that they are topologically transitive, and that the natural invariant measure, the so-called "Burger-Roblin measure", is ergodic, as soon as the geodesic flow admits a finite measure of maximal entropy, and this entropy is strictly greater than the codimension of the unipotent flow inside the maximal unipotent flow. The latter result generalises a theorem of Mohammadi and Oh.
- In the talk, I will present the main ideas of this work.
- 16h30-17h45 : Sean Lawton (George Mason University)
- Titre : Minimal generating sets for coordinate rings of representations
- Résumé : We will first define the moduli space of algebraic-group-valued representations of finitely presented groups. Then we will briefly describe how non-commutative rings influence the structure of the coordinate ring of these moduli spaces. Lastly, we will illustrate this general relationship by constructing minimal generating sets of the coordinate rings of these moduli spaces in some specific examples.
Mercredi 30 janvier 2019, Amphithéâtre Léon Motchane
- 14h30-15h45 : Juliette Bavard (CNRS & Université de Rennes 1)
- Titre : Two simultaneous actions of big mapping class groups
- Résumé : Mapping class groups of infinite type surfaces, also called "big" mapping class groups, arise naturally in several dynamical contexts, such as two-dimensional dynamics, one-dimensional complex dynamics, "Artinization" of Thompson groups, etc.
- In this talk, I will present recent objects and phenomena related to big mapping class groups. In particular, I will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. I will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). If time allows, I will explain how to use these simultaneous actions to construct nontrivial quasimorphisms on subgroups of big mapping class groups.
- This is joint work with Alden Walker.
- 16h30-17h45 : Daniel Monclair (Université Paris-Sud)
- Titre : Non-differentiability of limit sets in anti-de Sitter geometry
- Résumé : The study of Anosov representations deals with discrete subgroups of Lie groups that have a nice limit set, meaning that they share the dynamical properties of limit sets in hyperbolic geometry. However, the geometry of these limits sets is different: while limit sets in hyperbolic geometry have a fractal nature (e.g. non-integer Hausdorff dimension), some Anosov groups have a more regular limit set (e.g. C1 for Hitchin representations).
- My talk will focus on quasi-Fuchsian subgroups of SO(n,2), and show that the situation is intermediate: their limit sets are Lipschitz submanifolds, but not C1. I will discuss the two main steps of the proof. The first one classifies the possible Zariski closures of such groups. The second uses anti-de Sitter geometry in order to determine the limit cone of such a group with a C1 limit set.
- Based on joint work with Olivier Glorieux.
Lundi 10 décembre 2018, Amphithéâtre Léon Motchane
- 14h30-15h45 : William Goldman (University of Maryland)
- Titre : Dynamics and moduli of geometries on surfaces
- Résumé : We describe dynamical systems arising from the classification of locally homogeneous geometric structures on manifolds. Their classification mimics the classification of Riemann surfaces by the Riemann moduli space --- the quotient of Teichmüller space by the properly discontinuous action of the mapping class group. However, this action is misleading: mapping class groups generally act chaotically on character varieties. For fundamental examples, these varieties appear as affine cubics, and we relate the projective geometry of cubic surfaces to dynamical properties of the action.
- 16h30-17h45 : Samuel Tapie (Université de Nantes)
- Titre : Growth gap, amenability and coverings
- Résumé : Let Γ be a group acting by isometries on a proper metric space (X,d). The critical exponent δΓ(X) is a number which measures the complexity of this action. The critical exponent of a subgroup Γ'<Γ is hence smaller than the critical exponent of Γ. When does equality occur? It was shown in the 1980s by Brooks that if X is the real hyperbolic space, Γ' is a normal subgroup of Γ and Γ is convex-cocompact, then equality occurs if and only if Γ/Γ' is amenable. At the same time, Cohen and Grigorchuk proved an analogous result when Γ is a free group acting on its Cayley graph.
- When the action of Γ on X is not cocompact, showing that the equality of critical exponents is equivalent to the amenability of Γ/Γ' requires an additional assumption: a "growth gap at infinity". I will explain how, under this (optimal) assumption, we can generalize the result of Brooks to all groups Γ with a proper action on a Gromov hyperbolic space.
- Joint work with R. Coulon, R. Dougall and B. Schapira.
Lundi 19 novembre 2018, Amphithéâtre Léon Motchane
- 14h30-15h45 : Andrés Sambarino (CNRS & IMJ-PRG)
- Titre : Hausdorff dimension of a (stable) class of non-conformal attractors
- Résumé : The purpose of the talk is to explain a result in collaboration with B. Pozzetti and A. Wienhard expressing the Hausdorff dimension of certain attractors as a critical exponent. This class of attractors consists of limit sets of Anosov representations in PGLd (hence of non-conformal nature) that verify an extra open condition. If time permits, we will discuss implications of the formula to the geometry of the Hitchin component.
- 16h30-17h45 : Ferrán Valdez (UNAM & Max Planck Institute)
- Titre : Big mapping class groups
- Résumé : Mapping class groups associated to surfaces whose fundamental
group is not finitely generated are called "big", and their study is linked to classical problems in dynamics. In this talk, we discuss some of the basic properties of big mapping class groups, their simplicial actions, and how these can be used to prove that big mapping class groups "detect" surfaces or (if time allows) that the space of non-trivial quasimorphisms of a big mapping class group is infinite dimensional.
Lundi 8 octobre 2018, Amphithéâtre Léon Motchane
- 14h30-15h45 : Tobias Hartnick (Universität Giessen)
- Titre : Approximate lattices in nilpotent Lie groups
- Résumé : In order to analyze mathematical and physical systems it is often necessary to assume some form of order, e.g. perfect symmetry or complete randomness. Fortunately, nature seems to be biased towards such forms of order as well. During the second half of the 20th century, a new paradigm of "aperiodic order" was suggested. Instances of aperiodic order were discovered in different areas of mathematics, such as harmonic analysis and diophantine approximation (Meyer), tiling theory (Wang, Penrose) and additive combinatorics (Freiman, Erdös-Szemeredi); after some initial resistance is has now been accepted that aperiodic order also exists in nature in the form of quasicrystals.
- Together with Michael Björklund we have proposed a general mathematical framework for the study of aperiodic structures in metric spaces, based on the notion of an "approximate lattices". Roughly speaking, approximate lattices generalize lattices in the same way that approximate subgroups (in the sense of Tao) generalize subgroups. Approximate lattices in Euclidean space are essentially the "harmonious sets" of Meyer (a.k.a. mathematical quasi-crystals), but there are interesting examples in other geometries, such as symmetric spaces, Bruhat-Tits buildings or nilpotent Lie groups. It turns out that with every approximate lattice one can associate a dynamical system, which replaces the homogeneous space associated with a lattice - thus the study of approximate lattices can be considered as "geometric group theory enriched over dynamical systems".
- In this talk I will
(1) give an overview over the basic framework of approximate lattices and geometric approximate group theory;
(2) illustrate the framework by formulating Meyer's theory of harmonious sets in this language;
(3) time permitting, discuss some recent structure theory of approximate lattices in nilpotent Lie groups and applications to Bragg peaks in the Schrödinger spectrum of magnetic quasicrystals.
- Based on joint works with Michael Björklund (Chalmers), Matthew Cordes (ETH), Felix Pogorzelski (Leipzig) and Vera Tonić (Rijeka).
- 16h30-17h45 : Nicolas Bergeron (IMJ-PRG, Sorbonne Université)
- Titre : Eigenfunctions and random waves on locally symmetric spaces in the Benjamini-Schramm limit
- Résumé : I will consider the asymptotic behavior of eigenfunctions of the Laplacian on a compact locally symmetric manifold M "in the level aspect", that is as the volume of M tends to infinity. I will formulate a precise conjecture of "Berry type", and describe partial results obtained in a joint work with Miklos Abert and Étienne Le Masson.
Lundi 18 juin 2018, Amphithéâtre Léon Motchane
- 14h30-15h45 : Ilia Smilga (Yale University)
- Titre : Milnorian and non-Milnorian representations
- Résumé : In 1977, Milnor formulated the following conjecture: every discrete group of affine transformations acting properly on the affine space is virtually solvable. We now know that this statement is false; the current goal is to gain a better understanding of the counterexamples to this conjecture. Every group that violates this conjecture "lives" in a certain algebraic affine group, which can be specified by giving a linear group and a representation thereof. Representations that give rise to counterexamples are said to be non-Milnorian. We will talk about the progress made so far towards classification of these non-Milnorian representations.
- 16h30-17h45 : Brice Loustau (Rutgers University)
- Titre : Bi-Lagrangian structures and Teichmüller theory
- Résumé : A bi-Lagrangian structure on a manifold is the data of a symplectic form and a pair of transverse Lagrangian foliations. Equivalently, it can be defined as a para-Kähler structure, i.e. the para-complex analog of a Kähler structure. After discussing interesting features of bi-Lagrangian structures in the real and complex settings, I will show that the complexification of any Kähler manifold has a natural complex bi-Lagrangian structure. I will then specialize this discussion to moduli spaces of geometric structures on surfaces, which typically have a rich symplectic geometry. We will see that that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory while revealing new other features, and derive a few well-known results of Teichmüller theory. Time permitting, I will present the construction of an almost hyper-Kähler structure in the complexification of any Kähler manifold. This is joint work with Andy Sanders.
Lundi 14 mai 2018, Amphithéâtre Léon Motchane
- 14h30-15h45 : Maria Beatrice Pozzetti (Universität Heidelberg)
- Titre : Maximal representations on infinite dimensional symmetric spaces
- Résumé : An important application of bounded cohomology is the theory of maximal representations: a class of homomorphisms of fundamental groups of Kähler manifolds (most notably fundamental groups of surfaces and finite volume manifolds covered by complex hyperbolic spaces) in Hermitian Lie groups (such as Sp(2n,R) or SU(p,q)). These representations have striking geometric properties and, in some cases, are even superrigid. In my talk I will discuss a joint work with Bruno Duchesne and Jean Lécureux in which we study generalizations to actions on infinite dimensional Hermitian symmetric spaces.
- 16h30-17h45 : Genevieve Walsh (Tufts University & Chaire Jean Morlet, CIRM)
- Titre : Relatively hyperbolic groups with planar boundaries
- Résumé : We will first explain the concepts of relatively hyperbolic group and the Bowditch boundary. We will then give some interesting examples of groups whose boundaries embed in the two-sphere. The most prominent family of this type is the class of geometrically finite Kleinian groups. However, we show that there are lots of relatively hyperbolic groups with planar boundaries that are not virtually Kleinian. We formulate a conjecture about which groups with planar boundary are virtually Kleinian, and prove this in a certain case. This is joint work in progress with Chris Hruska.
Mercredi 25 avril 2018, Amphithéâtre Léon Motchane
- 14h30-15h45 : Jean-François Quint (CNRS & Université de Bordeaux)
- Titre : Complementary series
- Résumé : Complementary series are families of unitary representations of certain semisimple Lie groups and of groups of automorphisms of homogeneous trees. I will recall their definition for semisimple groups and explain how to prove their existence for groups of automorphisms of trees by a method which allows to extend this construction in order to build new representations of free groups.
- 16h30-17h45 : Daniele Alessandrini (Universität Heidelberg)
- Titre : Domains of discontinuity for (quasi-)Hitchin representations
- Résumé : Among representations of surface groups into Lie groups, the Anosov representations are the ones with the nicest dynamical properties.
- Guichard-Wienhard and Kapovich-Leeb-Porti have shown that their actions on generalized flag manifolds often admit co-compact domains of discontinuity, whose quotients are closed manifolds carrying interesting geometric structures.
- Dumas and Sanders studied the topology and the geometry of the quotient in the case of quasi-Hitchin representations (Anosov representations which are deformations of Hitchin representations). In a conjecture they ask whether these manifolds are homeomorphic to fiber bundles over the surface.
- In joint work with Qiongling Li, we can prove that the conjecture is true for (quasi-)Hitchin representations in SL(n,R) and SL(n,C), acting on projective spaces and partial flag manifolds parametrizing points and hyperplanes.
Lundi 19 mars 2018, Amphithéâtre Léon Motchane
- 14h30-15h45 : Anne Parreau (Université Grenoble Alpes)
- Titre : Geodesic currents, positive cross-ratios and degenerations of maximal representations
- Résumé : Degenerations of maximal representations of a surface group may be seen as maximal representations in Sp(2n,F) for some non-Archimedean real closed field F. We associate to every such maximal representation a geodesic current whose intersection number is the length function of the representation for the L1 norm. When the current is a measured lamination, we reconstruct an equivariant isometric embedding of the dual real tree in the Bruhat-Tits building of Sp(2n,F). This involves a general construction of an intersection current associated to a non necessarily continuous positive cross-ratio. This is joint work with Marc Burger, Alessandra Iozzi, and Beatrice Pozzetti.
- 16h30-17h45 : Charles Frances (Université de Strasbourg)
- Titre : Dynamics and topology on 3-dimensional Lorentz manifolds
- Résumé : A classical result of Myers and Steenrod states that the isometry group of a compact Riemannian manifold is a compact Lie transformation group. Also classical is the fact that this compactness property fails for general pseudo-Riemannian manifolds, allowing interesting dynamics for the group of isometries. In this talk, we will be interested by the topological, and dynamical consequences of the noncompactness of the isometry group. We will especially focus on the case of Lorentz manifolds, and we will present a complete topological classification of 3-dimensional closed Lorentz manifolds having a noncompact isometry group.
Lundi 12 février 2018, Amphithéâtre Léon Motchane
- 14h30-15h45 : Marc Burger (ETH Zürich)
- Titre : A structure theorem for geodesic currents and applications to compactifications of character varieties
- Résumé : We find a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the intersection function on the set of closed curves is bilipschitz equivalent to the length function with respect to a hyperbolic metric. We show that the subset of currents with positive systole is open and that the mapping class group acts properly discontinuously on it. As an application, we obtain in the case of compact surfaces a structure theorem on the length functions appearing in the length spectrum compactification both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This is joint work with Alessandra Iozzi, Anne Parreau, and Beatrice Pozzetti.
- 16h30-17h45 : Andrew Sanders (Universität Heidelberg)
- Titre : The topology of compact manifolds arising from Anosov representations
- Résumé : An Anosov representation of a word hyperbolic group Γ into a semisimple Lie group G is a dynamically defined strengthening of a quasi-isometric embedding of Γ into G, which serves as a flexible higher rank analogue of the notion of convex-cocompactness. In particular, Anosov representations yield interesting discrete subgroups of G. Guichard-Wienhard and Kapovich-Leeb-Porti constructed co-compact domains of proper discontinuity for these discrete subgroups lying in generalized flag manifolds G/P where P<G is a parabolic subgroup. Distinct domains of discontinuity are indexed by certain special subsets (ideals) in the Weyl group of G with respect to the Bruhat order. In this talk, we discuss the calculation of the homology groups of the quotient manifolds in the case when Γ is a closed surface group, and G is a complex simple Lie group. The formulas express the Betti numbers explicitly in terms of the combinatorial properties of the corresponding subset of the Weyl group of G. This yields a sufficient condition to distinguish the homotopy type of two quotient manifolds obtained from different ideals in the Weyl group. Time permitting, we will present some interesting special cases where the Poincaré polynomial can be expressed as a particularly simple rational function with the degree of the numerator and denominator depending on the genus of the surface.
Lundi 15 janvier 2018, Amphithéâtre Léon Motchane
- 14h30-15h45 : Hugo Parlier (Université du Luxembourg)
- Titre : Quantifying isospectral finiteness
- Résumé : Associated to a closed hyperbolic surface is its length spectrum, the set of the lengths of all of its closed geodesics. Two surfaces are said to be isospectral if they share the same length spectrum.
- The talk will be about the following questions and how they relate:
The approach to these questions will include finding adapted coordinate sets for moduli spaces and exploring McShane type identities.
- How many questions do you need to ask a length spectrum to determine it?
- How many different surfaces can be isospectral to a surface of a given genus?
- 16h30-17h45 : Sara Maloni (University of Virginia)
- Titre : Convex hulls of quasicircles in hyperbolic and anti-de Sitter space
- Résumé : Thurston conjectured that quasi-Fuchsian manifolds are uniquely determined by the induced hyperbolic metrics on the boundary of their convex core and Mess extended this conjecture to the context of globally hyperbolic anti de-Sitter spacetimes. In this talk I will discuss a universal version of Thurston and Mess' conjectures: any quasisymmetric homeomorphism from the circle to itself is obtained on the convex hull of a quasicircle in the boundary at infinity of the 3-dimensional hyperbolic (resp. anti-de Sitter) space. We will also discuss a similar result for convex domains bounded by surfaces of constant curvature K in (-1,0) in the hyperbolic setting and of curvature K in (-∞,-1) in the anti de-Sitter setting with a quasicircle as their asymptotic boundary.
- (This is joint work in progress with F. Bonsante, J. Danciger and J.-M. Schlenker.)
Lundi 4 décembre 2017, Amphithéâtre Léon Motchane
- 14h30-15h45 : Martin Deraux (Université Grenoble Alpes)
- Titre : Spherical CR structures on 3-manifolds
- Résumé : A spherical CR structure on a 3-manifold is a geometric structure modeled on the boundary at infinity of the complex hyperbolic plane, or in other words a (G,X)-structure with G=PU(2,1), X=S3. I will discuss spherical CR uniformizations, which are a special kind of spherical CR structure that arises by taking the manifold at infinity of a quotient of the ball by the action of a discrete group of isometries. I will explain how to construct some explicit uniformizations, including a 1-parameter family of (pairwise non-conjugate) spherical CR uniformizations of the figure eight knot complement.
- 16h30-17h45 : Matías Carrasco (Universidad de la República, Montevideo)
- Titre : Dimension drop of the harmonic measure of some hyperbolic random walks
- Résumé : We consider the simple random walk on two types of tilings of the hyperbolic plane. The first by 2π⁄q-angled regular polygons, and the second by the Voronoi tiling associated to a random discrete set of the hyperbolic plane, the Poisson point process. In the second case, we assume that there are on average λ points per unit area.
- In both cases the random walk (almost surely) escapes to infinity with positive speed, and thus converges to a point on the circle. The distribution of this limit point is called the harmonic measure of the walk.
- I will show that the Hausdorff dimension of the harmonic measure is strictly smaller than 1, for q sufficiently large in the Fuchsian case, and for λ sufficiently small in the Poisson case. In particular, the harmonic measure is singular with respect to the Lebesgue measure on the circle in these two cases.
- The proof is based on a Furstenberg type formula for the speed together with an upper bound for the Hausdorff dimension by the ratio between the entropy and the speed of the walk.
- This is joint work with P. Lessa and E. Paquette.
Lundi 16 octobre 2017, Amphithéâtre Léon Motchane
- 14h30-15h45 : Jean-Marc Schlenker (Université du Luxembourg)
- Titre : The renormalized volume of quasifuchsian manifolds
- Résumé : Quasifuchsian manifolds are an important class of hyperbolic 3-manifolds, classically parametrized by two copies of Teichmüller space. Their volume is infinite, but they have a well-defined finite "renormalized volume" which has nice properties, both analytic and "coarse". In particular, considered as a function over Teichmüller space, the renormalized volume provides a Kähler potential for the Weil-Petersson metric; moreover, it is within bounded additive constants of the volume of the convex core and is bounded from above by the Weil-Petersson distance between the conformal structures at infinity. After describing these properties, we will outline some recent applications (by Kojima, McShane, Brock, Bromberg, Bridgeman, and others) to the Weil-Petersson geometry of Teichmüller space or the geometry of hyperbolic 3-manifolds that fiber over the circle. We will then explain how properties of the renormalized volume suggest new questions and viewpoints on quasifuchsian manifolds. The talk will be accessible to nonexperts.
- 16h30-17h45 : Michelle Bucher (Université de Genève)
- Titre : Vanishing simplicial volume for certain affine manifolds
- Résumé : Affine manifolds, i.e. manifolds which admit charts given by affine transformations, remain mysterious by the very few explicit examples and their famous open conjectures: the Auslander Conjecture, the Chern Conjecture and the Markus Conjecture. I will discuss an intermediate conjecture, somehow between the Auslander Conjecture and the Chern Conjecture, predicting the vanishing of the simplicial volume of affine manifolds. In a joint work with Chris Connell and Jean-François Lafont, we prove the latter conjecture under some hypothesis, thus providing further evidence for the veracity of the Auslander and Chern Conjectures. To do so, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups in their fundamental group to have vanishing simplicial volume. This answers a special case of a question due to Lück. Joint with Chris Connell and Jean-François Lafont.
Lundi 19 juin 2017, Amphithéâtre Léon Motchane
- 14h30-15h45 : Kathryn Mann (University of California, Berkeley)
- Titre : Large scale geometry in large transformation groups
- Résumé : In this talk I will survey some recent work on coarse geometry of transformation groups, specifically, groups of homeomorphisms and diffeomorphisms of manifolds. Following a framework developed by C. Rosendal, many of these groups have a well defined quasi-isometry type (despite not being locally compact or compactly generated). This provides the right context to discuss geometric questions such as boundedness and subgroup distortion --- questions which have already been studied in the context of actions of finitely generated groups on manifolds.
- 16h30-17h45 : Camille Horbez (CNRS & Université Paris-Sud)
- Titre : Automorphismes de groupes hyperboliques et croissance
- Résumé : Soit G un groupe hyperbolique sans torsion, soit S une partie génératrice finie de G, et soit f un automorphisme de G. Nous cherchons à comprendre les taux de croissance possibles pour la longueur d'un élément g du groupe G (écrit comme un mot en les générateurs dans S) sous l'itération de f. Le cas où G est un groupe de surface ou un groupe libre est compris grâce à des résultats de Thurston et Bestvina-Handel. Nous montrons qu'en général, il n'y a qu'un nombre fini de taux de croissance exponentiels possibles lorsque l'élément g parcourt G. Par ailleurs, nous montrons la dichotomie suivante : tout élément a une croissance qui est soit exponentielle, soit polynomiale. Ceci est un travail en commun avec Rémi Coulon, Arnaud Hilion et Gilbert Levitt.
Lundi 22 mai 2017, Amphithéâtre Léon Motchane
- 14h30-15h45 : Olivier Guichard (Université de Strasbourg)
- Titre : L'adhérence de Zariski des représentations de Hitchin et des représentations positives
- Résumé : Nous montrerons que, pour de telles représentations ρ : π1Σ → G, l'adhérence de Zariski de ρ(Γ) contient le SL2 principal sauf lorsque Γ est un sous-groupe monogène de π1Σ, auquel cas cette adhérence de Zariski est un sous-groupe abélien régulier connexe. Ceci entraîne la classification des adhérences de Zariski possibles puisque la classification des groupes algébriques contenant le SL2 principal est connue. Enfin dans le cas où Γ = π1Σ (ou un sous-groupe d'indice fini) les paramètres de Hitchin déterminent l'adhérence de ρ(Γ).
- 16h30-17h45 : Jeffrey Danciger (University of Texas at Austin)
- Titre : Convex real projective structures and Anosov representations
- Résumé : We investigate the degree to which the geometry of a compact real projective manifold with boundary is reflected in the associated holonomy representation, a representation of the fundamental group in the projective general linear group PGL(n,R) which in general need not have any nice properties.
- We show that if the projective manifold is strictly convex, then its holonomy representation is projective Anosov, a condition which generalizes the dynamical properties of convex cocompact representations in rank one (e.g. hyperbolic) geometry. Conversely, a strictly convex projective manifold may be constructed from a projective Anosov representation that preserves a properly convex set in projective space. Applications include new examples of both convex projective manifolds and Anosov representations. Joint work with François Guéritaud and Fanny Kassel.
Lundi 24 avril 2017, Amphithéâtre Léon Motchane
- 14h30-15h45 : Rafael Potrie (Universidad de la República, Montevideo)
- Titre : Rigidité de régularité pour les représentations de Hitchin
- Résumé : Les représentations de Hitchin sont un exemple paradigmatique de représentations d'Anosov et ont été largement étudiées comme analogues en rang supérieur de l'espace de Teichmüller. Labourie a prouvé l'existence de courbes équivariantes dans l'espace des drapeaux complets. La trace de ces courbes dans l'espace projectif est toujours de classe C1 mais les courbes ne sont en général pas plus lisses que Hölder. Dans cet exposé on va donner une preuve du fait suivant : si la courbe est lisse alors la représentation est Fuchsienne. Les techniques sont aussi importantes pour des résultats de rigidité pour l'exposant critique. Cela fait partie d'un travail en collaboration avec A. Sambarino.
- 16h30-17h45 : Nicolas Tholozan (CNRS & ÉNS)
- Titre : Géométrie et dynamique des représentations maximales en rang 2
- Résumé : Parmi les représentations de groupes de surfaces à valeurs dans le groupe de Lie hermitien SO(2,n), celles dont l'invariant de Toledo est maximal forment une famille de représentations d'Anosov, dont les nombreuses propriétés géométriques et dynamiques ont été mises en évidence par les travaux de Labourie, Guichard et Wienhard.
Dans un travail en commun avec Brian Collier et Jérémy Toulisse, nous étudions plus en détail l'action de ces représentations sur les différents espaces homogènes de SO(2,n). Nous démontrons en particulier que l'exposant critique de ces représentations est majoré par 1, et qu'elles préservent une unique surface minimale dans l'espace symétrique de SO(2,n).
Lundi 13 mars 2017, Amphithéâtre Léon Motchane
- 14h30-15h45 : Bruno Klingler (Université Paris 7)
- Titre : La conjecture de Chern pour les variétés spéciales affines
- Résumé : Une variété affine (au sens de la géométrie différentielle) est une variété admettant un atlas de cartes à valeurs dans un espace affine V et à changements de cartes localement constants dans le groupe affine Aff(V). À la fin des années 50, Chern a conjecturé que la caractéristique d'Euler de toute variété affine compacte s'annule. Je discuterai cette conjecture, et sa preuve dans le cas où X est spéciale affine (i.e. X est affine et admet une forme volume parallèle).
- 16h30-17h45 : Jonathan Chaika (University of Utah et IHP)
- Titre : Cobounded foliations are a path connected subset of PMF
- Résumé : The space of projective measured foliations is (one of) the boundaries of Teichmüller space. One can consider a special subclass of this set that define Teichmüller geodesics whose projection to moduli space is contained in a compact set. These can be thought of as analogous to badly approximable rotations. The main result of the talk is that this set is path connected in high enough genus. This is joint work with Sebastian Hensel.
Lundi 6 février 2017, Amphithéâtre Léon Motchane
- 14h30-15h45 : Kasra Rafi (University of Toronto)
- Titre : Balls in Teichmüller space are not convex
- Résumé : We prove that when 3g − 3 + p > 3, the Teichmüller space of the closed surface of genus g with p punctures contains balls which are not convex in the Teichmüller metric. We analyze the quadratic differential associated to a Teichmüller geodesic and, as a key step, show that the extremal length of a curve (as a function of time) can have a local maximum. This is a joint work with Maxime Fortier Bourque.
- 16h30-17h45 : Arielle Leitner (Technion)
- Titre : Generalized cusps on convex projective manifolds
- Résumé : A convex projective manifold C = Ω/Γ is the quotient of convex subset of projective space, Ω, by a discrete group of projective transformations Γ ⊂ PGL(n+1,R). A generalized cusp in dimension 3 is a convex projective manifold that is the product of a ray and a torus. The holonomy centralizes a 1-parameter subgroup of PGL(n,R). I have shown: A generalized cusp on a properly convex projective 3-dimensional manifold is projectively equivalent to one of 4 possible cusps.
- For a generalized cusp C = Ω/Γ in dimension n, we require that ∂C is compact and strictly convex (contains no line segment) and that there is a diffeomorphism h : [0,∞) × ∂C → C. Together with Sam Ballas and Daryl Cooper we have classified generalized cusps in dimension n, and explored new geometries arising from such cusps. We show the holonomy of a generalized cusp is a lattice in one of a family of Lie groups G(λ) parameterized by a point λ = (λ1, ..., λn) ∈ Rn. More generally a maximal-rank cusp in a hyperbolic n-orbifold is determined by the similarity class of lattice in Isom(En−1).
Lundi 16 janvier 2017, Amphithéâtre Léon Motchane
- 14h30-15h45 : Yves Benoist (CNRS & Université Paris-Sud)
- Titre : Quasiisométries harmoniques
- Résumé : Dans un travail commun avec D. Hulin nous montrons que toute quasiisométrie entre variétés de Hadamard pincées est à distance bornée d'une unique application harmonique.
- 16h30-17h45 : Thierry Barbot (Université d'Avignon)
- Titre : Théorème de Pappus et représentations du groupe modulaire
- Résumé : Au début des années 90, R.E. Schwartz a montré que le théorème de Pappus permet de définir une famille d'actions du groupe modulaire sur le plan projectif avec des propriétés géométriques et dynamiques remarquables. Ces propriétés sont semblables à celles satisfaites par les représentations d'Anosov. Dans sa thèse (sous ma direction), V. Pardini Valerio a élucidé ce fait en montrant que les représentations de Schwartz, restreintes à un sous-groupe d'indice deux, sont limites de représentations d'Anosov.
- Je présenterai ce travail de thèse, et les progrès récents obtenus conjointement avec Gye-Seon Lee.
Lundi 12 décembre 2016, Amphithéâtre Léon Motchane
- 14h30-15h45 : François Ledrappier (CNRS & LPMA, Université Paris 6)
- Titre : Théorème limite local en courbure négative
- Résumé : On considère le noyau de la chaleur p(t,x,y) sur le revêtement universel d'une variété compacte de courbure négative et on en donne un équivalent quand t tend vers l'infini. La démonstration introduit une nouvelle famille équivariante naturelle de mesures à l'infini, liée au bas du spectre du Laplacien λ0 sur le revêtement universel. Il s'agit d'un travail en commun avec Seonhee Lim.
- L'exposé commencera par une introduction au sujet, de type colloquium.
- 16h20-17h35 : Gilles Courtois (CNRS & IMJ-PRG, Université Paris 6)
- Titre : Rigidité horosphérique des variétés hyperboliques
- Résumé : Je parlerai du théorème suivant. On considère une variété compacte M de dimension supérieure ou égale à 3 et de courbure négative. Si une horosphère de M est plate, alors M est de courbure constante. Il s'agit d'un travail en commun avec Gérard Besson et Sa'ar Hersonsky.
Lundi 14 novembre 2016, Amphithéâtre Léon Motchane
- 14h30-15h45 : François Labourie (Université de Nice-Sophia Antipolis)
- Titre : Groupes de surfaces dans les réseaux
- Résumé : Il s'agit d'un travail en commun avec Jeremy Kahn et Shahar Mozes. Nous montrons que les réseaux dans certains groupes de Lie non compacts -- en particulier tous les groupes complexes -- possèdent des sous-groupes isomorphes à des groupes de surfaces. Nous montrons de plus que ces sous-groupes sont "quasi-symétriques" par rapport à un choix préalable d'un SL(2) en un sens à préciser. Nous donnerons quelques idées de la preuve, qui suit le schéma du travail fondateur de Kahn-Markovic, en insistant sur le nouvel outil : l'étude de triangles dans les variétés de drapeaux.
- 16h30-17h45 : Virginie Charette (Université de Sherbrooke)
- Titre : Les tores d'Einstein dans l'univers d'Einstein de dimension trois
- Résumé : Nous discuterons de tores d'Einstein et de leurs intersections, à l'aide de la correspondance entre l'univers d'Einstein de dimension trois et l'espace des plans lagrangiens dans R^4 symplectique. Comme application, nous discuterons d'un critère pour que des surfaces appelées "surfaces croches" soient disjointes, en reliant ce critère à celui de Danciger-Guéritaud-Kassel.
- Travail conjoint avec Jean-Philippe Burelle, Dominik Francoeur et Bill Goldman.
Lundi 26 septembre 2016, Amphithéâtre Léon Motchane
- 14h45-16h : Anna Wienhard (Universität Heidelberg)
- Titre : Positivity and higher Teichmüller theory (pdf)
- Résumé : Classical Teichmüller space describes the space of conformal structures on a given topological surface S. It plays an important role in several areas of mathematics as well as in theoretical physics.
- Higher Teichmüller theory generalizes several aspects of classical Teichmüller theory to the context of Lie groups of higher rank, such as the symplectic group PSp(2n,R) or the special linear group PSL(n,R). So far, two families of higher Teichmüller spaces are known. The Hitchin component, which is defined when the Lie group is a split real forms, and the space of maximal representations, which is defined for Lie groups of Hermitian type. Interestingly, both families are linked with various notions of positivity in Lie groups.
- In this talk I will give an introduction to higher Teichmüller theory, introduce new positive structures on Lie groups and discuss the (partly conjectural) relation between the two.
- 16h30-17h45 : Bertrand Deroin (CNRS)
- Titre : Représentations super-maximales des groupes de sphères épointées à valeurs dans PSL(2,R)
- Résumé : On présentera une classe particulière de représentations des groupes des sphères épointées dans PSL(2,R) que nous appelons super-maximales. On montrera que ces représentations sont totalement non hyperboliques, dans le sens que les courbes fermées simples sont envoyées sur des éléments elliptiques ou paraboliques. On montrera également que les représentations super-maximales sont géométrisables par des orbifolds hyperboliques dans un sens très fort. Enfin, on montrera que les représentations super-maximales définissent des composantes compactes dans certaines variétés de caractères relatives, qui sont symplectomorphes à des espaces projectifs complexes, ce qui généralise un résultat de Benedetto-Goldman dans le cas des sphères moins quatre points. Il s'agit d'un travail en collaboration avec Nicolas Tholozan.