Dominating surface group representations by Fuchsian ones

We prove that a representation from the fundamental group of a closed surface of negative Euler characteristic with values in the isometry group of a Riemannian manifold of sectional curvature bounded by -1 can be dominated by a Fuchsian representation. Moreover, we prove that the domination can be made strict, unless the representation is discrete and faithful in restriction to an invariant totally geodesic 2-plane of curvature -1. When applied to representations into PSL(2,R) of non-extremal Euler class, our result is a step forward in understanding the space of closed anti-de Sitter 3-manifolds.Comment: Added details in lemma 2.3. Corrected a mistake about the link with Toledo's theorem. Removed a superfluous assumption in theorem F and added a last section about "perspectives in higher rank

Random conformal dynamical systems

We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversaly conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudo-group. We prove that either there exists a measure invariant under all the elements of the group (or the pseudo-group), or almost surely a long composition of maps contracts exponentially a ball. We deduce some results about the unique ergodicity.Comment: 61 page

A transfer principle: from periods to isoperiodic foliations

In this paper we study the dynamics and topology of the isoperiodic foliation defined on the Hodge bundle over the moduli space of genus $g\geq 2$ curves

The oriented graph of multi-graftings in the Fuchsian case

We prove the connectedness and calculate the diameter of the oriented graph of graftings associated to exotic complex projective structures on a compact surface S with a given holonomy representation of Fuchsian type. The oriented graph of graftings is the graph whose vertices are the equivalence classes of marked CP^1-structures on S with a given fixed holonomy, and there is an oriented edge between two structures if the second is obtained from the first by grafting.Comment: Improved version. The paper chaged title: from "The oriented graph of graftings..." to "The oriented graph of multi-graftings...