Convegno
“MOVING TO HIGHER RANK: FROM HYPERBOLIC TO ANOSOV”

I will present a relative version of Gromov's Vanishing Theorem about amenable open covers with small multiplicity

By the Nielsen-Thurston classification theorem, there are three types of elements in the mapping class group of a surface: periodic, reducible, and pseudo-Anosov. We describe an efficient (i.e., polynomial-time) algorithm to distinguish pseudo-Anosov mapping classes

We will discuss a correlation theorem for pairs of locally Hölder continuous potentials with strong entropy gaps at infinity on a topologically mixing countable Markov shift with the BIP property. This extends a result of Lalley on shifts of finite type, and we will explain its application to the dynamics of (pairs of) Hitchin representations of a punctured surface. This talk is based on joint work in progress with Lien-Yung Nyima Kao.

We provide two examples of convex cocompact Kleinian groups whose limit set is a Pontryagin sphere. The examples are in dimension 4 and 6 respectively, and are obtained from reflection groups.

A discrete and faithful representation of a surface group in PSL(2,C) is said to be quasi-Fuchsian when it preserves a Jordan curve on the Riemann sphere. Classically these objects lie at the intersection of several areas of mathematics and have been studied (for example) using complex dynamics, Teichmüller theory, and 3-dimensional hyperbolic geometry. From a dynamical perspective, an important invariant of such representations is the Hausdorff dimension of the invariant Jordan curves (typically a very fractal object). It is elementary to see that this number is always at least 1. A celebrated result of Bowen establishes it is equal to 1 if and only if the quasi-Fuchsian representation is Fuchsian, that is, it is conjugate in PSL(2,R). I will first describe this classical picture and then report on recent joint work with James Farre and Beatrice Pozzetti where we prove a generalization of Bowen's result for the much larger class of hyperconvex representations of surface groups in PSL(d,C) (where d is arbitrary).

In this talk we will describe a generalisation of the Gordon-Luecke Theorem that says that knots in the 3-sphere are determined by their complements. We will show that the same holds when the ambient manifold is a circle bundle.