Convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space

We construct convex-cocompact representations of fundamental groups of closed hyperbolic surfaces into the isometry group of the infinite-dimensional hyperbolic space using bendings. We prove that convex-cocompact representations of finitely generated groups in the group of isometries of the infinite-dimensional hyperbolic space form an open set in the space of representations and that the space of deformations (up to conjugation) obtained by bending an irreducible representation of a surface group is infinite-dimensional.