Workshop
Teichmüller theory and surfaces in 3-manifolds

CRM De Giorgi, Pisa, 9 - 13 June 2014

Welcome Schedule Participants Intensive period (26 May - 20 June)

Webpage hosted by Centro De Giorgi Registration and financial support Location and accommodation

---

All the talks will take place at Palazzo del Castelletto (view the map).

Timetable

Invited speakers:

· Jeff Brock (Brown University), On the geometry of random hyperbolic 3-manifolds. Abstract.

With respect to various notions of randomness, one can ask what topological and geometric behaviors of a 3-manifold obtain `generically'. ÊIn this talk I will discuss a proof of a conjecture of N. Dunfield and W. Thurston that the volume of a random Heegaard splitting grows linearly in the word-length of the gluing map as an element of the mapping class group. ÊThis talk represents joint work with Igor Rivin and Juan Souto.

· Bertrand Deroin (ENS Paris), TBA. Abstract.

· Guillame Dreyer (Notre Dame), Parametrizing Hitchin components. Abstract.

Let S be a closed, connected, oriented surface of negative Euler characteristic. For n>2, Hitchin components Hit_n(S) are components of the PSL_n(R)-character variety R_{PSL_n(R)}(S)$ that correspond to Teichmüller components T(S) in the case where n=2. Over the recent years, groundbreaking work has revealed fundamental geometric, dynamical and algebraic properties for the representations in Hit_n(S)$. In particular, these Hitchin representations turn out to share many features with the classic Fuchsian representations. In a joint work with Francis Bonahon, we construct a geometric, real analytic parametrization of the Hitchin components Hit_n(S) that it is based on topological choices only. The construction strongly relies on two independent approaches to studying Hitchin representations: the dynamical approach of Anosov representation, introduced by F. Labourie; and the algebraic-combinatorial approach of Positive representation, developed by V. Fock and A. Goncharov. In essence, given a maximal geodesic lamination lambda in S, our parametrization is an extension of Thurston's shear coordinates along the leaves of lambda on the Teichmüller space T(S), combined with Fock-Goncharov's coordinates on the moduli space of positive framed local systems of a punctured surface.

· David Dumas (UIC), Computing the image of Thurston's skinning map. Abstract.

Thurston's skinning map is a holomorphic map between Teichmueller spaces that arises in the construction of hyperbolic structures on compact 3-manifolds. ÊI will describe the theory and implementation of a method for numerically computing the images of skinning maps in some low-dimensional examples. ÊThe key to the method is that each point in the image of the skinning map represents an intersection between two Lagrangian subvarieties of the SL(2,C) character variety of a surface group. Ê The skinning image is computed by tracking the movement of these intersections as one of the varieties (the Bers slice) moves in a holomorphic family. ÊThis is joint work with Richard Kent.

· François Fillastre (Cergy-Pontoise), Convex sets in Minkowski space. Abstract.

From works of G. Mess and F. Bonsante, we know that flat space-times with compact hyperbolic Cauchy surface can be isometrically embedded as convex domains in Minkowski space. Those convex domains belong to a general class of convex sets, among which they have the property to be invariant under subgroups of isometries. Those general convex sets can be studied using tools from the classical theory of convex bodies. In particular, we can define area measures, and look at the problem of prescribing a measure in two relevant cases, that leads to analog of the Christoffel problem (joint work with Giona Veronelli) and of the Minkowski problem (joint work with Francesco Bonsante).

· Stefano Francaviglia (Bologna), Moduli spaces of (branched) projective structures. Abstract.

In this talk we will discuss some properties of the moduli spaces of (possibly) branched {CP}^1-structures on surfaces. Examples of such structures are hyperbolic metric with cone-singularities multiple of 2\pi and translation surfaces. We will focus in local/global surgeries that allow to travel in the moduli space of such structures. (Joint work with G. Calsamiglia and B. Deroin.)

· William Goldman (College Park), Flat affine and projective manifolds. Abstract.

This talk will survey the few known results and open problems concerning affine and projective structures. I will concentrate on examples.

· Colin Guillarmou (Cergy-Pontoise), The renormalized volume of geometrically finite hyperbolic 3-manifolds. Abstract.

We discuss how to define the renormalized volume on a geometrically finite hyperbolic 3 manifold, in particular in the case where there are rank-1 cusps. We also consider limits of the renormalized volume as a function on the the classical Schottky space when we approach the boundary. (Joint work with S. Moroianu and F. Rochon.)

· Ursula Hamenstädt (Bonn), Constructing surface subgroups in locally symmetric spaces. Abstract.

We give a geometric version of a construction of Kahn and Markovic of surface subgroups in cocompact lattices in $SL(2,C)$. We use this to extend the construction to cocompact lattices in rank one Lie groups and explain its significance to an understanding of the mapping class group of a closed surface of genus at least two.

· Zheng Huang (CUNY), Closed minimal surfaces in quasi-Fuchsian manifolds. Abstract.

It is well-known that each quasi-Fuchsian manifold contains at least one closed incompressible minimal surface. An intriguing question is how many? Understanding these minimal surfaces turns out to have many applications in Teichmuller theory, hyperbolic three-manifolds and other fields. In joint work with Biao Wang, we partially answer above question. A very short answer is the supremum over all quasi-Fuchsian manifolds of all genera at least two is positive infinity.

· Alessandra Iozzi (ETH Zürich), TBA. Abstract.

· Fanny Kassel (Lille), Complete constant-curvature spacetimes in dimension 3.  Abstract.

The Minkowski space R^{2,1} is the Lorentzian analogue of the Euclidean space R^3; the anti-de Sitter space AdS^3 is the Lorentzian analogue of the hyperbolic space H^3. I will survey some recent results on the geometry and topology of their quotients by discrete groups, with strong links with two-dimensional hyperbolic geometry. In particular, I will explain how the quotients of R^{2,1} by free groups (Margulis spacetimes) are « infinitesimal versions » of quotients of AdS^3. This is joint work with J. Danciger and F. Guéritaud. 

· François Labourie (Paris XI), TBA. Abstract.

· Christopher Leininger (UIUC), Entropy and homology for pseudo-Anosov homeomorphisms. Abstract.

By analyzing surfaces in 3-manifolds, we deduce an interesting connection between the entropy of pseudo-Anosov homeomorphisms and their action on homology. This is joint work with I. Agol and D. Margalit.

· M. Beatrice Pozzetti (ETH Zürich), Maximal representations of complex hyperbolic lattices in SU(m,n). Abstract.

Let G be a lattice in SU(1,p). Maximal representations of G in SU(m,n) are those homomorphisms that maximize the generalized Toledo invariant, a cohomological invariant that extends the well studied Toledo invariant for representations of fundamental groups of surfaces into Lie groups of Hermitian type. We show that, if p is greater than one, interesting rigidity phenomena appear: the only Zariski dense maximal representation of G in SU(m,n), with n greater than m, is the lattice embedding in SU(1,p). This allows to prove that the restriction to G of the diagonal embedding of SU(1,p) in SU(m,pm+k) is locally rigid.

· Kasra Rafi (Toronto), Teichmuller space is semi-hyperbolic. Abstract.

We provide a bi-combing of Teichmuller space by modifying the Teichmuller geodesic paths. As corollary, we conclude that the Dehn functions of the Teichmuller space are Euclidean.

· Andrès Sambarino (Paris XI), The Pressure metric for convex representations. Abstract.

The purpose of the talk is to describe a Riemannian metric on the Hitchin component, analogous to the Weil-Petersson metric on Teichmuller space. This is a joint work with M. Bridgeman, R. Canary and F. Laoburie.

· Andrew Sanders (U. Illinois), A new proof of Bowen's theorem on Hausdorff dimension of quasi-circles . Abstract.

A quasi-Fuchsian group is a discrete group of Mobius transformations of the Riemann sphere which is isomorphic to the fundamental group of a compact surface and acts properly on the complement of a Jordan curve: the limit set. In 1979, Bowen proved a remarkable rigidity theorem on the Hausdorff dimension of the limit set of a quasi-Fuchsian group: it is equal to 1 if and only if the limit set is a round circle. This theorem now has many generalizations. We will present a new proof of Bowen's result as a by-product of a new lower bound on the Hausdorff dimension of the limit set of a quasi-Fuchsian group. This lower bound is in terms of the differential geometric data of an immersed, incompressible minimal surface in the quotient manifold. If time permits, generalizations of this result to other convex co-compact surface groups will be presented.

· Maxime Wolff (Paris VI), The modular action on PSL(2,R)-characters in genus two Abstract.

I will present an ongoing joint work with Julien Marche, in which we describe the dynamics of the mapping class group action on the PSL(2,R) character variety in genus two.

---