Sphere-packing and volume in hyperbolic 3-space

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Abstract
A hyperbolic 3-manifold is a Riemannian manifold of constant sectional curvature -1. We will restrict our attention to complete orientable hyperbolic 3-manifolds M; as such, we can think of M as H3/F where F is a discrete torsion-free subgroup of Isom÷ (H3), the orientation-preserving isometries of hyperbolic 3-space. We will generally work in the upper-half-space model H 3 of hyperbolic 3-space, in which case PGL(2, C) acts as orientation-preserving isometries on n 3 by extending the action of PGL(2, C) on the Riemann sphere (boundary of H 3) to H 3. An orbifold is a space locally modelled on R" modulo a finite group action. Complete orientable hyperbolic 3-orbifolds Q correspond to discrete subgroups F of PGL(2, C). If the discrete group F corresponding to M or Q has parabolic elements then M or Q is said to be cusped. Unless otherwise stated, we will assume all manifolds and orbifolds are orientable. Mostow's theorem implies that a complete, hyperbolic structure on a 3-orbifold of finite volume is unique. Consequently, hyperbolic volume is a topological invariant for orbifolds admitting such structures. J0rgensen and Thurston proved (see IT] section 6.6) that the set of volumes of complete hyperbolic 3-manifolds is well-ordered and of order type m '°. 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 Vo,. Further, all volumes of closed manifolds are isolated, while volumes of cusped manifolds are limits from below (thus the notation Vo,). Modifying the proofs in the J0rgensen-Thurston theory yields similar results for complete hyperbolic 3-orbifoids (this result is folklore, and we will not prove it here). In particular, there is a hyperbolic 3-orbifold of minimum volume V'~, and a cusped hyperbolic 3-orbifold of minimum volume V~.

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