Thurston's Double Limit Theorem provided a criterion ensuring convergence, up to subsequence, of a sequence of quasifuchsian representations. This criterion was the key step in his proof that 3-manifolds which fiber over the circle are geometrizable.  In this talk, we describe a complete characterization of when a sequence of quasifuchsian representations has a convergent subsequence.  Moreover, we will see that the asymptotic behavior of the conformal structures determines the ending laminations and parabolic loci of the algebraic limit and how the algebraic limit ``wraps'' inside the geometric limit. (The results described are joint work with Jeff Brock, Ken Bromberg, Cyril Lecuire and Yair Minsky.)

The Hitchin component is a (special) connected component of the space of homomorphisms of a surface group into $\textrm{PSL}(d,\mathbb{R}).$  This component is a higher rank analogue of the Teichmuller space of the surface.  The purpose of the talk is to show that the critical exponent of a Hitchin representation has a rigid upper bound. This is a joint work with Rafael Potrie.

There is a well-known result known as the collar lemma for hyperbolic surfaces. It has the following consequence: 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 one can choose on the surface. Furthermore, this lower bound for the length of a grows logarithmically as we shrink the length of b. By slightly weakening this lower bound, we generalize this statement to hold for all Hitchin representations instead of just hyperbolic structures. This is joint work with Gye-Seon Lee.

There are natural incidence structures on the boundary of the complex hyperbolic space and on some suitable boundary S associated to the group PU(m,n). Such structures have striking rigidity properties: I will prove that a (measurable) map from the boundary of the complex hyperbolic space to S that preserves these incidence structures needs to be algebraic. This implies that, if G is a lattice in SU(1,p) and n is greater than m, there exist Zariski dense maximal representations of G in SU(m,n) only if (m,n) is equal to (1,p). In particular the restriction to G of the diagonal embedding of SU(1,p) in SU(m,pm+k) is locally rigid.



 

We show that the entropy of a pseudo-Anosov which fixes a k-dimensional subspace of homology of a genus g surface is comparable to (k+1)/g. This answers a question of Ellenberg. Key use is made of a recent inequality of Kojima-McShane giving an upper bound on volume in terms of dilatation

We exhibit some dynamical properties of the Weil Petersson geodesic flow. As an application, we obtain geometric information on random hyperbolic 3-manifolds which fibre over the circle.

We discuss the asymptotic behavior of the number of closed geodesics of a given combinatorial type on a hyperbolic surface (the closed curve can be disconnected or have self intersections).

We will see that this question is closely related to the distribution of lengths of closed curves on a random pants decomposition of a hyperbolic surface. We use the ergodicity of the earthquake flow to study this problem.

More generally, we consider the action of mapping class group on the space of geodesic currents, and discuss the growth of the orbit of an arbitrary point. We discuss both topological and geometric versions of this problem.