Winter School on Geometric Structures in Nice
Université de Nice, Laboratoire Jean-Alexandre Dieudonné.
January 7-11, 2019.

## Program

• Nicolas Bergeron: Arithmetics and Hyperbolic Manifolds
In these lectures I will explain the construction of arithmetic hyperbolic manifolds. I will then focus on their homology, free part and then torsion part. Finally, I will explain why 3 dimensional arithmetic manifolds are good candidates for being « topological expanders » and what it is good for.
Exercises
• Ken Bromberg: Renormalized volume of hyperbolic 3-manifolds
I will begin with some basics on deformations of Riemannian metrics with a special emphasis on hyperbolic metrics in dimensions 2 and 3. Then I will describe some of the basic properties of convex co-compact hyperbolic 3-manfiolds and their renormalized volume. In the final lecture I will survey some recent results and open problems.
Exercises
• Indira Chatterji: $CAT(0)$ Cubical Complexes
We will discuss CAT(0) cubical complexes and spaces with walls. We shall see how one can construct a cubulation, that is a proper cocompact action on a CAT(0) cubical complex, from a space with walls. We will also discuss Bergeron-Wise boundary criterion for cubulation, that allowed them to build on results by Kahn-Markovic to cubulate fundamental groups of closed hyperbolic 3 manifolds.
Exercises
• David Dumas: Introduction to Opers
Opers are a class of geometric structures on a compact Riemann surface $X$ that are associated with a reductive complex Lie group $G$. In these lectures we will discuss the general theory and the construction of some examples, focusing on the case $G=\mathrm{SL}(n,\mathbb{C})$. Here, an oper can be seen in several equivalent guises: As a rank-$n$ holomorphic vector bundle over $X$ with a connection (satisfying certain axioms), as an immersion of the universal cover of $X$ in the projective space $\mathbb{CP}^{n-1}$ (satisfying a transversality condition), or as a linear ordinary differential operator of order $n$ acting on a line bundle over $X$. Will we discuss the equivalences between these different perspectives, the identification between opers for $\mathrm{SL}(2,\mathbb{C})$ and complex projective structures, and the parameterization of the space of $\mathrm{SL}(n,\mathbb{C})$-opers by tuples of holomorphic differentials on $X$.
Exercises
• Alessandra Iozzi: A dual interpretation of Gromov's proof of Mostow Rigidity theorem as in Thurston's notes
I will present a rigidity theorem for representations of the fundamental group of a hyperbolic n-manifold into SO(1,n). This gives as a corollary a different interpretation of a proof of Mostow Rigidity theorem, for which we basically replace smearing and straightening by bounded cohomology. Indeed the proof will be preceded by a gentle introduction to continuous bounded cohomology in its most useful reincarnation.
Exercises