Abstract
Consider in hyperbolic space H-3 the one parameter family of immersed (non embedded) constant mean curvature surfaces of revolution D-tau with constant mean curvature H > 1. The parameter tau is an element of (-infinity, 0) is the analogue of the " necksize" of the Delaunay surfaces in Euclidean space. It is proved that when tau -> -infinity, there exists a branch of surfaces with constant mean curvature H which bifurcate from D-tau. Furthermore, we prove that these new surfaces have only a discrete group of symmetries. The proof consists in a detailed study of the behaviour of the eigenvalues of the Jacobi operator when tau tends to -infinity, together the bifurcation theorem of Crandall-Rabinowitz.