Robert Meyerhoff

Minimum volume cusped hyperbolic three-manifolds

This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic 3-manifold with volume <= 2.848 can be obtained by a Dehn filling on one of 21 cusped hyperbolic 3-manifolds. We also show how this result can be used to construct a complete list of all one-cusped hyperbolic three-manifolds with volume <= 2.848 and all closed hyperbolic three-manifolds with volume <= 0.943. In particular, the Weeks manifold is the unique smallest volume closed orientable hyperbolic 3-manifold.

The cusped hyperbolic 3-orbifold of minimum volume

An orbifold is a space locally modelled on R 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/Y, where T is a discrete subgroup of Isom+(if), the orientation-preserving isometries of hyperbolic 3-space. An orientable hyperbolic 3-manifold corresponds to a discrete, torsion-free subgroup of Isom+(i/). We will work in the upper-half-space model H of hyperbolic 3-space, in which case PGL(2, C) acts as isometries on H by extending the action of PGL(2, C) on the Riemann sphere (boundary of H) to H. 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. Mostows 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;. 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

MOM TECHNOLOGY AND VOLUMES OF HYPERBOLIC 3-MANIFOLDS

This paper is the first in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Here we introduce Mom technology and enumerate the hyperbolic Mom-n manifolds for n ≤ 4. Our longterm goal is to show that all low-volume closed and cusped hyperbolic 3-manifolds are obtained by filling a hyperbolic Mom-n manifold, n ≤ 4 and to enumerate the low-volume manifolds obtained by filling such a Mom-n. William Thurston has long promoted the idea that volume is a good measure of the complexity of a hyperbolic 3-manifold (see, for example, [Th1] page 6.48). Among known low-volume manifolds, Jeff Weeks ([We]) and independently Sergei Matveev and Anatoly Fomenko ([MF]) have observed that there is a close connection between the volume of closed hyperbolic 3-manifolds and combinatorial complexity. One goal of this project is to explain this phenomenon, which is summarized by the following: Hyperbolic Complexity Conjecture 0.1. (Thurston, Weeks, Matveev-Fomenko) The complete low-volume hyperbolic 3-manifolds can be obtained by filling cusped hyperbolic 3-manifolds of small topological complexity. Remark 0.2. Part of the challenge of this conjecture is to clarify the undefined adjectives low and small. In the late 1970’s, Troels Jorgensen proved that for any positive constant C there is a finite collection of cusped hyperbolic 3-manifolds from which all complete hyperbolic 3-manifolds of volume less than or equal to C can be obtained by Dehn filling. Our long-term goal stated above would constitute a concrete and satisfying realization of Jorgensen’s Theorem for “low” values of C. A special case of the Hyperbolic Complexity Conjecture is the long-standing Smallest Hyperbolic Manifold Conjecture 0.3. The Weeks Manifold MW, obtained by (5,1), (5,2) surgery on the two components of the Whitehead Link, is the unique oriented hyperbolic 3-manifold of minimum volume. Note that the volume of MW is 0.942.... All manifolds in this paper will be orientable and all hyperbolic structures are complete. We call a compact manifold hyperbolic if its interior supports a complete hyperbolic structure of finite volume.

Hyperbolic 3-manifolds

Hyperbolic geometry is the non-Euclidean geometry discovered by Janos Bolyai, N. I. Lobachevsky, and…