Abstract : We give a geometric classi cation of regular ends with constant mean curvature 1 and nite total curvature,embedded in hyperbolic space. We prove each such end is either asymptotic to a catenoid cousin or else is asymptoticto a horosphere. We also study some symmetry properties of constant mean curvature 1 surfaces in hyperbolic spaceassociated to minimal surfaces in Euclidean space. We describe the constant mean curvature 1 surfaces in H 3 associated(isometric) to the family of surfaces in R 3 isometric to the helicoid. Key words: hyperbolic space, constant mean curvature 1 surfaces, Bryant representation, regular ends, associatedsurfaces. Math. Classification: 53C42IntroductionThe theory of constant mean curvature 1 surfaces in hyperbolic space was created byRobert Bryant in a pioneering paper [Br]. There, Bryant gives a holomorphic parametrizationof such surfaces (simply connected) which is a close analogue of the Weierstrass representationfor minimal surfaces in R 3 . Later on, Umehara and Yamada, following Bryant’s idea have givenimportant contributions to the theory. Particularly, they introduced the notion of regular ends(see Definition 2-1) and also they provided some new examples and techniques to constructcomplete constant mean curvature 1 surfaces with regular and non-regular ends (see [U-Y,1],[U-Y,2], [U-Y,3]).In this paper we prove that if