J. DIFFERENTIAL GEOMETRY 21 (1985) 231 268 3 DIMENSIONAL LORENTZ SPACE FORMS AND SEIFERT FIBER SPACES RAVI S. KULKARNI & FRANK RAYMOND Table of Contents §1. Introduction 231 §2. Special features of the 3 dimensional Lorentz geometry of constant curvature 1 235 §3. Subgroups of PSL 2 (R) and PSL 2 (R) 239 §4. Subgroups of /oίSbo), the level of a subgroup 243 §5. Properly discontinuous subgroups of / 0 (^i) 245 §6. Space forms with solvable fundamental groups 247 §7. Finiteness of Level 248 §8. Standard Lorentz orbifolds and Seifert orbifolds 252 §9. Topology of standard orbifolds 258 §10. Homogeneous space forms 264 §11. A strange space form 265 References 267 1. Introduction A space form is a complete pseudo Riemannian manifold of dimension > 2 with constant curvature. A Lorentz space form is a space form with a Lorentz metric of signature + — . In this paper we study 3 dimensional Lorentz space forms of constant curvature 1, and unless there is a possibility of confusion, these will be often referred to simply as space forms. The standard linear model for this geometry (the "3 dimensional anti de Sitter space") is S 1 ' 2 = {(*, y)\x 9 y e R 2 , |x| 2 \y\ 2 = l} * O(2,2)/O(l,2), cf. [38, p. 334]. This set up differs markedly from the usual Riemannian set ups in two respects: (1) the isotropy subgroup 0(1,2) is noncompact, so 0(2,2) does not act properly on S ιa . This feature substantially restricts the discrete subgroups of 0(2,2) which can act properly discontinuously on S 1 ' 2 . (2) On Received November 28, 1983 and, in revised form, January 10, 1985. The first author was partially supported by an NSF grant and a Guggenheim fellowship, and the second author by an NSF grant.