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Exposé 1161 in the Séminaire N. Bourbaki

Related project(s):
28Rigidity, deformations and limits of maximal representations

We study the Thurston–Parreau boundary both of the Hitchinand of the maximal character varieties and determine therein an open setof discontinuity for the action of the mapping class group. This result isobtained as consequence of a canonical decomposition of a geodesic cur-rent on a surface of finite type arising from a topological decompositionof the surface along special geodesics. We show that each componenteither is associated to a measured lamination or has positive systole. Fora current with positive systole, we show that the intersection function onthe set of closed curves is bi-Lipschitz equivalent to the length functionwith respect to a hyperbolic metric.

Related project(s):
28Rigidity, deformations and limits of maximal representations

In this paper we investigate the Hausdorff dimension of limitsetsof Anosov representations. In this context we revisit and extend the frameworkof hyperconvex representations and establish a convergence property for them,analogue to a differentiability property. As an applicationof this convergence,we prove that the Hausdorff dimension of the limit set of a hyperconvex rep-resentation is equal to a suitably chosen critical exponent. In the appendix, incollaboration with M. Bridgeman, we extend a classical result on the Hessianof the Hausdorff dimension on purely imaginary directions.

Related project(s):
28Rigidity, deformations and limits of maximal representations

We study Anosov representation for which the image of the bound-ary map is the graph of a Lipschitz function, and show that theorbit growthrate with respect to an explicit linear function, the unstable Jacobian, is inte-gral. Several applications to the orbit growth rate in the symmetric space areprovided.

Related project(s):
28Rigidity, deformations and limits of maximal representations

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite dimensional totally geodesic subspace on which the action is maximal. In the opposite direction we construct examples of geometrically dense maximal representation in the infinite dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, that we are able to construct in low ranks or under some suitable Zariski-density assumption, circumventing the lack of local compactness in the infinite dimensional setting.

Related project(s):
28Rigidity, deformations and limits of maximal representations