invent, math. 87, 441-456 (1987) ///venZ/o//e$ mathematicae 9 Springer-Verlag 1987 On universal groups and three-manifolds H.M. Hilden 1, M.T. Lozano 2, J.M. Montesinos 2, 3,, and W.C. Whitten 4 1 University of Hawaii, Honolulu, Hawaii, USA 2 Facultad de Ciencias, Universidad de Zaragoza, Spain 3 MSRI, Berkeley,California, USA 4 University of Southwestern La, Lafayette, Louisiana, USA In his notes [Th2] Thurston described a hyperbolic three-orbifold B 3 whose singular locus is the Borromean rings and whose isotropy groups are all cyclic of order four. Thus, this orbifold is defined by a finitely generated group U of orientation-preserving isometries of hyperbolic three-space IH 3. In this paper we prove that U is universal, i.e., for every closed and oriented three-manifold M 3 there exists a subgroup G N U of finite index such that M 3 is homeomorphic to the quotient IH3/G. In other words, M 3 is a hyperbolic orbifold covering B 3. Every subgroup G< U of finite index defines a closed, oriented three-ma- nifold ]I-I3/G whose fundamental group is G/F, where F is the subgroup of G generated by the elements of finite order. Therefore, the simply connected three-manifolds correspond precisely to the subgroups GNU of finite index that are generated by elements of finite order. This offers a new approach to the Poincar6 Conjecture and we have considered it worthwhile to obtain a system of generators of U N PSL(2, C). The generators given in this paper are three matrices whose entries lie in a subring of the ring of algebraic integers. The main result of this paper is a consequence of the result that every closed, oriented three-manifold M 3 is a covering S 3 branched over the Bor- romean rings and having branching indices 1, 2 and 4. The first part of this result (that the Borromean rings are universal) was first proved in [HLM1] ; here we sharpen it by adding the qualification about the branching indices. We remark that the proof of the universality of the Borromean rings given here is different from the one given in [HLM1]. This paper is self-contained and can be read without reference to the earlier papers on universal knots and links, i.e., the germinalpaper by Thurston [Thl] and [HLM1, HLM2, HLM3]. This paper is organized as follows. In Sect. 1 we show that the Borromean rings are universal with branching indices 1, 2 and 4. In Sect. 2 we construct the associated regular covering (branched over the Borromean rings) and show that its branching indices are all 4. In Sect. 3 we describe a well-known * Supported by Comit6 Conjunto Hispano-Norteamericano and in part by NSF Grant 8120790