February 28th - March 9th // 2023
Topics on the Geometry of Locally Symmetric Spaces
at Montevideo
A school for Master and Ph. D. students consisting on five 6h-mini-courses, each one treating different aspects of the subject. Participation is open for anyone interested.
Mini-courses
Hyperbolic 3-manifolds in the spirit of Thurston
We will introduce the geometric theory of hyperbolic 3-manifolds with a focus on convex cocompact hyperbolic 3-manifolds, i.e. hyperbolic 3-manifolds which contain a compact convex submanifold. We will give alternative characterizations of convex cocompact hyperbolic 3-manifolds from dynamical and other viewpoints.
We will see that the boundary of the convex core is a pleated surface and discuss some of the general theory of pleated surfaces in hyperbolic 3-manifolds. As time permits we will sketch Thurston's proof that 3-manifolds which fiber over the circle (with pseudo-Anosov monodromy) admit hyperbolic structures.
Minimal entropy of hyperbolic manifolds
The minimal entropy of a closed manifold M is the infimum, over the set of all riemannian metrics g on M of volume 1, of the volume entropy of (M,g). When M carries a hyperbolic metric, the minimal entropy of M is achieved exactly at the hyperbolic metric.
The goal of the course is to describe the proof of this property, due to Katok in dimension 2, and to Besson-Courtois-Gallot in dimension at least 3. The proof in dimension 2 uses conformal geometry and dynamical properties of the geodesic flow and the higher dimension case relies on a geometric construction of a map F:(M,g) → (M,hyp) with good volume contraction properties.
Geometry and dynamics of symmetric spaces of noncompact type
We will explore the interaction between the geometry, dynamics and algebra of rank one and (especially) higher rank symmetric spaces of noncompact type when viewed as nonpositively curved manifolds. Special attention will be paid to the ideal boundary and its metric and spherical building structures, and how these relate to the dynamics of the geodesic and Weyl chamber flows and group actions. Along the way we will discuss some of the properties of discrete groups of isometries acting on such spaces.
Counting closed geodesics on hyperbolic surfaces à la Mirzakhani
In her thesis Maryam Mirzakhani established the asymptotic growth, as L tends to infinity, of the number of simple closed geodesics of length at most L in a hyperbolic surface. In particular, she proved that for any hyperbolic surface S of genus g we have that the limit
exists and is positive. Some years later, using very different methods, she generalized this result to hold also for closed geodesics with self-intersections.
In this mini-course I will give an idea of how to prove these results. In fact, we will give a different proof from the original ones, using simpler methods, which allows us to treat both cases at once. Our main tools will be geodesic currents (which can be seen as the completion of the set of closed geodesics) and train tracks (combinatorial models of simple geodesics on surfaces). Along the way we will also discuss hyperbolic surfaces and Teichmuller space, measured laminations and the Thurston measure, and the Thurston compactification of Teichmuller space using geodesic currents.