# Recent Advances in Surface Group Representations

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## Speakers

• Daniele Alessandrini
• Thierry Barbot
• Gye-Seon Lee
• Qiongling Li
• Beatrice Pozzetti
• Andrés Sambarino
• Nicolas Tholozan
• Tengren Zhang

## Participants

• Vincent Alberge
• Shinpei Baba
• Brian Collier
• Charles Frances
• Fanny Kassel
• Slava Kharlamov
• Dmitriy Slutskiy
• Florian Stecker

## Schedule

Monday Tuesday Wednesday Thursday Friday
9:30 Zhang 9:30 Li 9:30 Pozzetti 9:30 Pozetti
11:00 Lee 11:00 Barbot 11:00 Fock 11:00 Alessandrini
14:00 Zhang 14:00 Pozzetti 12:15 Sambarino 14:00 Zhang
15:30 Tholozan 15:30 Sambarino 15:30 Sambarino

## Dinner

A dinner will be held on Tuesday evening.

## Titles and Abstracts

• Daniele Alessandrini, Geometric structures on 3-manifolds and Higgs bundles

Higgs bundles can be used to construct geometric structures on manifolds. I will explain how to use them to construct the closed Anti-de Sitter 3-manifolds, and to understand some of their properties. This is joint work with Qiongling Li.

• Thierry Barbot, Fuchsian polygonal surfaces in Minkowski and the decorated Teichmüller space

In the paper "Fuchsian polyhedra in Lorentzian space-forms" [Mathematische Annalen 350, 2011] F. Fillastre proved that any singular Euclidean metric with with cone angles bigger than $$2\pi$$ on a closed surface can be uniquely realized (up to global isometry) as a convex polygonal surface in a flat globally hyperbolic spacetime whose holonomy group is linear. In this talk, I will mention the work under progress of L. Brunswic aiming to extend this result in the case of flat spacetimes containing massive or BTZ particles. I will present the connection with the Decorated Teichmüller space ("The Decorated Teichmüller space of punctured surfaces", Commun. Math. Phys. 113, 1987).

• Vladimir Fock, Abelianization of the character varieties

• Gye-Seon Lee Collar lemma for Hitchin representations

There is a classical result first due to Keen known as the collar lemma for hyperbolic surfaces. A consequence of the collar lemma is that if two closed curves A and B on a closed orientable hyperbolizable surface have non-zero geometric intersection number, then there is an explicit lower bound for the length of A in terms of the length of B, which holds for any hyperbolic structure on the surface. By slightly weakening this lower bound, we generalize this statement to hold for all Hitchin representations. This is a joint work with Tengren Zhang.

• Qiongling Li, Anosov property of tensor products of representations

• Beatrice Pozzetti, Maximal representations and buildings

I will describe geometric properties of the action on affine buildings associated to unbounded sequences of maximal representations, focusing on elements with fixed points. I will also discuss a version of the Collar Lemma for maximal representations and for limiting actions. This is a joint work with Marc Burger

• Andrés Sambarino, Anosov representations: the thermodynamical viewpoint

On this series of talks we will focus on the construction of the Pressure metric for spaces of Anosov representations (joint with Bridgeman, Canary and Labourie) and a dynamical characterization of the Zariski closure of a Hitchin representation (joint with Potrie).

• Nicolas Tholozan, Length spectrum of convex projective structures on surfaces

By work of Loftin and Labourie, Hitchin representations in $$\mathrm{PSL}(3,\mathbb{R})$$ are parametrized by pairs $$(X,\Phi)$$, where $$X$$ is a Riemann surface and $$\Phi$$ a holomorphic cubic differential on $$X$$. Here we prove that the length spectrum of the Hitchin representation associated to $$(X,\Phi)$$ is uniformly bigger than the length spectrum of the Fuchsian representation associated to $$X$$. Our proof uses the interpretation of these Hitchin representations as convex projective structures, and relies on a comparison result between the Blaschke and Hilbert metrics on a proper convex domain.

• Tengren Zhang, The limit curve for Hitchin representations

There are two main goals for these three talks. The first is to describe an analog of the Fenchel-Nielsen coordinates on the Hitchin component. The second is to use the first to explain some results about how Hitchin representations degenerate when we perform a type of deformation which I call "deforming along internal sequences". An important tool in these results is a theorem obtained by combining the work of Labourie and Guichard, which says that Hitchin representations are exactly the representations that preserve a Frenet curve in the space of complete flags.