Produits d'isométries d'un espace symétrique de rang un et compacité à la Mumford-Mahler

E. Falbel and R. Wentworth

If G is the isometry group of a symmetric space with negative curvature, we prove the properness of the map which associates to a pair elements of G in fixed conjugacy classes the conjugacy class of the product. It follows that for G=PU(2,1) every conjugacy class of loxodromic elements can be realized as the product of a pair of loxodromics in specific classes. We also consider the space of embeddings in G of fundamental groups of closed surfaces. When the minimal translation distance of elements in the image of the homomorphisms is bounded from below by a positive constant, the space is compact modulo the action of the mapping class group and conjugaison. This gives a generalization of theorems of Mumford and Thurston in the cases G=PSL(2,R) and PSL(2,C).