The cusped hyperbolic 3-orbifold of minimum volume

Abstract

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) 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].)