Dynamics and geometry of discrete subgroups of Lie groups

In his celebrated thesis, Margulis obtained precise estimates on the number of closed geodesics on any compact negatively curved Riemannian manifold. To achieve this, he constructed a finite flow-invariant measure on the unit tangent bundle which satisfies the mixing property. Later, Patterson and Sullivan found another, elegant and elementary, way to construct the mixing measure. Their construction was adapted to many different geometrical settings, such as CAT(-1) geometry in Roblin's work. In this talk, we adapt the construction to yet another geometrical setting: a discrete group of projective transformations preserving a properly convex open subset of the projective space and its Hilbert metric.

I'll discuss a Besson-Courtois-Gallot type rigidity theorem for convex real projective manifolds endowed with the Hilbert metric in dimension at least three. Applications include a universal lower bound on the Hilbert volume of a manifold. This is based on joint work with Dave Constantine and Ilesanmi Adeboye. 

Let $G$ be a connected semisimple real linear Lie group, $P$ be its minimal parabolic subgroup, and $N$ be the unipotent radical of $P$. We give a precise description of all nontrivial $N$-invariant ergodic and $P^\circ$-quasi-invariant Radon measures on $\Gamma\backslash G$ where $P^\circ$ is the connected component of $P$ containing $e$ and $\Gamma<G$ is a Zariski dense Anosov subgroup with respect to $P$. This is joint work with Hee Oh.

Discrete subgroups of semisimple Lie groups such as SL(d,R) are objects rich in geometry and dynamics. Convex cocompact subgroups of rank-one Lie groups such as SL(2,R) are an especially nice class of discrete hyperbolic subgroups with good geometric and dynamical properties. Geometrically finite subgroups of rank-one Lie groups are a slightly larger class of discrete subgroups which allow for certain controlled failures of hyperbolicity while keeping relatively good geometric and dynamical properties. 

I will introduce analogous class of (relatively) hyperbolic discrete subgroups of higher-rank Lie groups, such as SL(d,R) with d at least 3: Anosov subgroups, first defined by Labourie and Guichard--Wienhard, and relatively Anosov subgroups.

The space of representations from the fundamental group of a closed surface into PSL(n,R) up to PSL(n,R)-conjugation contains a distinguished connected component: the Hitchin component. The goal of this talk is to introduce this component and its real spectrum compactification.

Part I discusses the relevant concepts of compactifications and real algebraic geometry.

Part II gives a characterisation of points in the real spectrum compactification of the Hitchin component by defining F-positive and F-Hitchin representations.

Hitchin components are natural generalizations of the classical Teichmüller space. In the setting of SL(3,R), the Hitchin component parameterizes the holonomies of convex real projective structures. By studying Blaschke metrics, which are Riemannian metrics associated to such structures, along with their limits, we obtain a compactification of the SL(3,R) Hitchin component. We show the boundary objects are hybrid structures, which are in part flat metric and in part laminar. These hybrid objects are natural generalizations of measured laminations, which are the boundary objects in Thurston's compactification of Teichmüller space. (joint work with Andrea Tamburelli)

In this talk we will discuss the energy functionals on Teichmüller space that are associated to Hitchin representations. We will show that these functionals are strictly plurisubharmonic functions on Teichmüller space. After that we consider the question whether a Hitchin representation is uniquely determined by its energy functional. We answer a similar question in a simpler setting (non-positively curved metrics on surfaces) and discuss work in progress on how these results can be expanded to include Hitchin representations.

Abstract Part 1: Cataclysms on Teichmüller space

Teichmüller space is the space of hyperbolic structures on a surface and has a rich structure. In particular, it admits a set of coordinates, called "shearing coordinates", that have been introduced by Thurston and Bonahon. These coordinates are closely related to deformations of hyperbolic surfaces. In this talk, I will introduce these deformations, namely earthquakes and cataclysms, together with geodesic laminations and transverse cycles, which are needed to define the deformations.

Abstract Part 2: Cataclysms for Anosov representations

Teichmüller space can be seen as a subset of the representation variety from the fundamental group of S into PSL(2,R). Replacing PSL(2,R) with a semi simple Lie group G, we can look at so-called Anosov representations. They share some properties with Teichmüller space, and one can ask: Is there an analogue of cataclysm deformations for Anosov representations? The answer to this question is yes. In this talk, I will explain what is important to know about Anosov representations in our setting and give an idea of the construction of cataclysms.

In part 1 I discuss undistorted subgroups of isometries of hyperbolic space and their local-to-global principle. In part 2 I discuss a characterization of Anosov representations due to Kapovich-Leeb-Porti that directly generalizes the undistortion condition. They defined Morse quasigeodesics in higher rank symmetric spaces and proved a suitable local-to-global principle. The talk concludes with applications of a quantified version of the local-to-global principle.

Anosov representations from a hyperbolic group Gamma to a semisimple Lie group G are dynamically well-behaved discrete representations. They form an open subset of the representation space. Anosov representations are extremely useful, but the exact set of them is known only in few examples, usually those where it fills out a whole connected component. For G=SL(3,R) and Gamma a triangle reflection group, we classify all Anosov representations. This includes one component with Anosov and non-Anosov representations. Other than the more well-studied Hitchin representations, the limit curves of these Anosov representations do not bound a convex domain.

Anosov representations were introduced by Labourie for fundamental groups of closed negatively curved Riemannian manifolds and further generalized by Guichard-Wienhard for more general Gromov hyperbolic groups. They form a rich and stable class of discrete subgroups of Lie groups with special dynamical and geometric properties. In this talk, we are going to exhibit examples of quasi-isometric embeddings of hyperbolic groups into SL(d,R) (d greater or equal than 7) which fail to be algebraic limits of Anosov representations into SL(d,R).