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In this paper we survey some recent results on rank one symmetric space.

In this paper we analyze the limit set of nonelementary subgroups acting by isometries on the product of two pinched Hadamard manifolds. Following M. Burger’s and P. Albuquerque’s works, we study the properties of Patterson-Sullivan’s measures on the limit sets of graph groups associated to convex cocompact groups.

We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $SU(p,q)$ (resp. $S{O}^{*}\left(2n\right)$, $n$ odd) and the surface group is maximal in some $S\left(U\right(p,p)\times U(q-p\left)\right)\subset SU(p,q)$ (resp. $S{O}^{*}(2n-2)\times SO\left(2\right)\subset S{O}^{*}\left(2n\right)$). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. García-Prada and P. Gothen.