Conference on Geometric Structures in Nice
Université de Nice, Laboratoire Jean-Alexandre Dieudonné.
January 14-18, 2019.

## Program

• Jean-Philippe Burelle: $P$-Anosov representations of free groups in $\text{Sp}(2n,\mathbb{R})$
Guichard-Wienhard recently generalized the notion of Lusztig positivity in order to define new higher Teichmueller spaces. The symplectic group $\text{Sp}(2n,\mathbb{R})$ admits two different kinds of positive structures, one in the space of complete isotropic flags and one in the Lagrangian Grassmannian. I will explain how each type of positivity can be used to define natural classes of Schottky groups which are $P$-Anosov representations of the free group. These generalize previous constructions of Charette-Francoeur-Lareau-Dussault and Burelle-Treib. We show that every $P$-Anosov representation can be continuously deformed to a Schottky group in this sense, and study the space of Schottky groups. This is work in progress with Fanny Kassel and Virginie Charette.

• Valentina Disarlo: Generalized stretch lines for surfaces with boundary
We will discuss some natural generalizations of Thurston's distance for surfaces with boundary, in particular the arc metric. We will construct a large family of geodesics for the Teichmueller space of surfaces with boundary with respect to the arc metric, which we will call " generalized stretch lines". We will prove that the Teichmueller space with the arc metric is a geodesic metric space, and that it is a Finsler space. This generalizes a result by Thurston on punctured surfaces. This is joint work with Daniele Alessandrini (University of Heidelberg).

• Federica Fanoni: Curve graphs for infinite-type surfaces
For surfaces of finite-type, studying the action of the mapping class group on a graph, called curve graph, has proved very useful to understand properties of the group itself. In the case of infinite-type surfaces (e.g. surfaces of infinite genus), the classical curve graph is not interesting from the coarse geometry viewpoint. I will discuss why and when we can (or can't) construct interesting graphs in the infinite-type case. Joint work with Matt Durham and Nick Vlamis.

• Selim Ghazouani: Renormalisation in one-dimensional dynamics and geometric structures on surfaces
Building upon work of Herman, Yoccoz, Khanin-Sinai from the 80s, I will try to give some motivations for questions concerning the generic behaviour of one-dimensional dynamical systems, such as circle diffeormorphims with break points or more generally generalised interval exchange transformations.
In turns out that in many cases moduli spaces of such dynamical systems bear a lot more structure than one would initially expect. We will try to explain how this extra structure comes from the connexion between such dynamical systems and corresponding geometric structures on surfaces.
This connection allows for an interpretation of certain renormalisation operators as actions of the mapping class group.
We will try to illustrate all these considerations with two examples where this approach yields somewhat significant dynamical results. The corresponding structures are complex affine structures on compact surfaces and representations of the free group in $\text{PSL}(2,\mathbb{R})$ respectively.

• Elise Goujard: Masur-Veech volumes of principal strata, intersection numbers and hyperbolic geometry
I will present a formula for the Masur-Veech volumes of the principal strata of half-translation surfaces (as well as Siegel-Veech constants) in terms of intersection numbers involving psi-classes. This formula closely relates the counting of square-tiled surfaces of fixed combinaotiral type with Mirzakhani's counting of simple closed geodesic multicurves on hyperbolic surfaces, and leads to several conjectures for the large genus asymptotics. This is a joint work with V.Delecroix, P.Zograf and A.Zorich.

• François Guéritaud: Geometric transition and canonical triangulations for the punctured torus
There are many parallels between the isometry groups of hyperbolic 3-space and of anti-de Sitter (2+1)-space, its Lorentzian counterpart. In particular, quasifuchsian groups (limit curves, convex cores, bending laminations) can be defined in both settings, and there is a continuous way of "transitioning" from one to the other. We will show that in the case of punctured torus groups, a natural "Delaunay decomposition" of the convex core (also related to the bending) is well-behaved under this transition. This is joint work with Sara Maloni.

• Richard Kenyon: Dimers and Circle Patterns
A circle pattern is a way to draw a planar graph so that each face is cyclic. The dimer model is a probability measure on the set of dimer covers (perfect matchings) of a planar graph. There is a surprising link between the dimer model on a bipartite planar graph and an associated circle pattern. In fact for a given bipartite graph there is an essentially bijective correspondence between circle patterns with embedded dual and dimer measures.

• Qiongling Li: Projective structures with holonomy in (Quasi-)Hitchin representations
In this talk, we study the topology of manifold admitting projective structures with holonomy in (Quasi)-Hitchin representations of surface groups into the Lie group $\text{SL}(n,\mathbb{R})$ or $\text{SL}(n,\mathbb{C})$. In particular, we focus on representations in a small neighborhood of Fuchsian locus inside the representation variety of $\text{SL}(n,\mathbb{R})/\text{SL}(n,\mathbb{C})$. This is joint work with Daniele Alessandrini.

• Sara Maloni: Convex hulls of quasicircles in hyperbolic and anti-de Sitter space
Thurston conjectured that quasi-Fuchsian manifolds are determined by the induced hyperbolic metrics on the boundary of their convex core and Mess generalized those conjectures to the context of globally hyperbolic AdS spacetimes. In this talk I will discuss a generalization of these conjecture to convex hulls of quasicircles in the boundary at infinity of hyperbolic and anti-de Sitter space. (This is joint work in progress with Bonsante, Danciger and Schlenker.)

• Julien Maubon: Rigidity of maximal representations of complex hyperbolic lattices
Maximal representations of surface groups in Hermitian semisimple Lie groups form a very interesting class of representations.
Here we focus on the higher dimensional case, that is on maximal representations of cocompact lattices $\Gamma$ of ${\rm SU}(n,1)$, with $n \geq 2$.
I will explain that such maximal representations are strongly rigid and can be completely classified.
First, they exists if and only if every simple factor of the target Hermitian Lie group $G$ is isomorphic to ${\rm SU}(p,q)$ for some $(p,q)$ with $p\geq nq$, or possibly, if $n=2$, to the exceptional group ${\rm E}_6$. Moreover. they are essentially induced by a Lie group homomorphism from ${\rm SU}(n,1)$ to $G$.
This is a joint work with Vincent Koziarz.

• Hugo Parlier: Lengths of geodesics and moduli spaces
This talk will be about geodesics on hyperbolic surfaces and identities involving their lengths. These identities are equations which relate lengths of certain geodesics and which remain true over entire moduli spaces. The talk will be about an ongoing project with Ara Basmajian and Ser Tan Peow where we study a family of identities relating lengths of orthogeodesics that interpolate between identities of Basmajian and McShane, and which hold true for surfaces with geodesic, cusp or cone-angle boundary.

• Anne Parreau: Boundary of higher Teichmuller spaces and geodesic currents
Geodesic currents were introduced by Bonahon to study Thurston's compactification of the Teichmuller space. In higher rank, this compactification can be generalized replacing the length function by the Jordan projection, seen as a Weyl chamber valued length function. Labourie, Martone and Zhang have constructed geodesic currents associated to hitchin or maximal representations. We will explain how this can be generalized to the boundary and allow to understand some structure of the degenerations of such representations. In particular we construct a natural non-empty open domain of discontinuity for the action of the mapping class group on the boundary. This is joint work with Marc Burger, Alessandra Iozzi, and Beatrice Pozzetti.

• Joan Porti: A local characterization of Anosov representations
Let $X=G/K$ be a symmetric space of noncompact type and $\Gamma$ a hyperbolic group. For a representation of $\Gamma$ in $G$, I give a characterization for this representation to be Anosov in terms of the image of finitely many elements of $\Gamma$. This is joint work with M. Kapovich and B. Leeb

• Beatrice Pozzetti: Critical exponent and Hausdorff dimension for Anosov representations
Whenever $G$ is a convex cocompact subgroup of the group of isometries of the hyperbolic space, Patterson-Sullivan theory allows to relate the asymptotic growth rate of orbit points for the action of $G$ on $\mathbb{H}^n$ and the Hausdorff dimension of the limit set of $G$ on the boundary at infinity. The relation between the Hausdorff dimension of the limit set of an Anosov representation and a suitable orbit growth rate is much more elusive since, on the one hand, the action of a higher rank group on its boundaries is not conformal, and, on the other, many different orbit growth functions can be considered. In my talk I'll report on joint work with A. Sambarino and A. Wienhard in which we find large classes of Anosov representations for which we can obtain such a relation.

• Richard Schwartz: Iterated Barycentric subdivision
The barycentric subdivision of a simplex can be iterated, producing an infinite family of simplices. I'll explain why the set of these simplices is dense in 2,3, and 4 dimensions. The 2 dimensional case, due to Barany, Beardon and Carne, and the 3 dimensional case, are pretty easy. I needed a computer-assisted proof for the 4 dimensional case.

• Andrea Seppi: Examples of four-dimensional geometric transition
Roughly speaking, a geometric transition is a deformation of geometric structures on a manifold, by "transitioning" between different geometries. Danciger introduced a new such transition, which enables to deform from hyperbolic structures to Anti-de Sitter structure, going through another type of real projective structures called "half-pipe", and provided conditions for a compact 3-manifold to admit a geometric transition of this type. By extending a construction of Kerckhoff and Storm, I will describe examples of finite-volume geometric transition in dimension 4. This is joint work with Stefano Riolo.

• Ilia Smilga: Milnorian and non-Milnorian representations
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.

• Jérémy Toulisse: Compact connected components of relative character varieties
In a recent paper, Bertrand Deroin and Nicolas Tholozan introduced the notion of supra-maximal representations of the fundamental group of the punctured sphere into $\text{PSL}(2,\mathbb{R})$. These representations have many surprising properties and form compact connected components of the relative character variety. In this talk, I will explain how the theory of parabolic Higgs bundles gives a way to construct similar representations into more general Hermitian Lie groups like $\text{SU}(p,q)$. The corresponding components are compact and isomorphic to certain quiver varieties. This is joint work with Nicolas Tholozan.

• Tengren Zhang: Affine actions with Hitchin linear part
We prove that if a surface group acts properly on $\mathbb{R}^d$ via affine transformations, then its linear part is not the lift of a $\text{PSL}(d,\mathbb{R})$-Hitchin representation. To do this, we proved two theorems that are of independent interest. First, we showed that $\text{PSO}(n,n)$-Hitchin representations, when viewed as representations into $\text{PSL}(2n,\mathbb{R})$, are never Anosov with respect to the stabilizer of the $n$-plane. Following Danciger-Gueritaud-Kassel, we also view affine actions on $\mathbb{R}^{n,n-1}$ as a geometric limit of isometric actions on $\mathbb{H}^{n,n-1}$. The second theorem we prove is a criterion for when an affine action on $\mathbb{R}^{n,n-1}$ is proper, in terms of the isometric actions on $\mathbb{H}^{n,n-1}$ that converge to it. This is joint work with Jeff Danciger, with some overlap with independent work by Sourav Ghosh.