The cusped hyperbolic 3-orbifold of minimum volume
Bulletin of the American Mathematical Society
An orbifold is a space locally modelled on R n modulo a finite group action. We will restrict our attention to complete orientable hyperbolic 3-orbifolds Q\ thus, we can think of Q as H 3 /Y, where T is a discrete subgroup of Isom+(if 3 ), the orientation-preserving isometries of hyperbolic 3-space. An orientable hyperbolic 3-manifold corresponds to a discrete, torsion-free subgroup of Isom+(i/ 3 ). We will work in the upper-half-space model H 3 of hyperbolic 3-space, in which case PGL(2, C)
... h case PGL(2, C) acts as isometries on H 3 by extending the action of PGL(2, C) on the Riemann sphere (boundary of H 3 ) to H 3 . If the discrete group T corresponding to Q has parabolic elements, then Q is said to be cusped. (For more details on this paragraph see [T, Chapter 13].) Unless otherwise stated, we will assume all manifolds and orbifolds are orientable. Mostow's theorem implies that a complete, hyperbolic structure of finite volume on a 3-orbifold is unique. Consequently, hyperbolic volume is a topological invariant for orbifolds admitting such structures. J0rgensen and Thurston proved (see [T, §6.6]) that the set of volumes of complete hyperbolic 3-manifolds is well-ordered and of order type u; w . In particular, there is a complete hyperbolic 3-manifold of minimum volume V\ among all complete hyperbolic 3-manifolds and a cusped hyperbolic 3-manifold of minimum volume K,. Further, all volumes of closed manifolds are isolated, while volumes of cusped manifolds are limits from below (thus the notation V^). Modifying the proofs in the J0rgensen-Thurston theory yields similar results for complete hyperbolic 3-orbifolds (but see the remark at the end of this paper). In particular, there is a hyperbolic 3-orbifold of minimum volume, and a cusped hyperbolic 3-orbifold of minimum volume. We prove THEOREM. Let Qi = H 3 /Yi where Ti = PGL(2, Ö3) and 0 3 =ring of integers in Q(\/-3). The orbifold Q\ has minimum volume among all orientable cusped hyperbolic S-orbifolds. Note. Qi is the orientable double-cover of the (nonorientable) tetrahedral orbifold with Coxeter diagram o-o-011& (see [T, Theorem 13.5.4] and [H, §1]). This tetrahedral orbifold has fundamental domain 1/24 of the ideal regular hyperbolic tetrahedron (use the symmetries). In particular, Q\ has a cusp and its volume is 1/12 the volume of the ideal regular tetrahedron T, i.e. vol(Qi) = VI12 « 0.0846, where V = vol(T).