In this manuscript, we give an overview of the tools and techniques needed for successfully classifying “low-complexity” Kleinian groups. In particular, we focus on extracting topological and geometric properties of discrete Kleinian groups, such as bounds on tube radii, cusp geometry, volume, relators in group presentation, and similar quantities. A key point of this manuscript is to explain how a discrete set of solutions (or their closure) can be found using continuous methods, in particular by searching over a continuous parameter space of groups. These methods provide an effective avenue for studying and classifying hyperbolic 3-manifolds that satisfy some geometric or topological constraints.

In this note, we answer a combinatorial question that is inspired by cusp geometry of hyperbolic 3-manifolds. A table-top necklace is a collection of sequentially tangent beads (i.e. spheres) with disjoint interiors lying on a flat table (i.e. a plane) such that each bead is of diameter at most one and is tangent to the table. We analyze the possible configurations of a necklace with at most 8 beads linking around two other spheres whose diameter is exactly 1. We show that all the beads are forced to have diameter one, the two linked spheres are tangent, and that each bead must be tangent to at least one of the two linked spheres. In fact, there is a 1-parameter family of distinct configurations.

We classify the complete hyperbolic 3-manifolds admitting a maximal cusp of volume at most 2.62. We use this to show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non-hyperbolic fillings; to show that the figure-8 knot complement and its sister are the unique hyperbolic 3-manifolds with minimal volume maximal cusps; and to extend results on determining low volume closed and cusped hyperbolic 3-manifolds.

We prove two conjectures of C. Gordon. We show that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10, and that the maximal intersection number between exceptional slopes is 8. The proof uses a combination of new geometric techniques and a rigorous computer-assisted calculation.

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.

This is an expository paper on Mom-technology, describing the recent work of the authors in this area (found in arXiv:math/0606072, arXiv:0705.4325, and arXiv:0809.0346) concerning the use of Mom-technology to find the minimum-volume compact hyperbolic 3-manifold and the 10 smallest cusped hyperbolic 3-manifolds. In addition we provide a survey of a selection of results on volumes of hyperbolic 3-manifolds, and a discussion of outstanding open problems in this area.

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.

Let M be a 3–manifold whose boundary consists of tori. The computer program SnapPea [20], created by Jeff Weeks, can approximate whether or not M is a complete hyperbolic manifold. However, until now, there has been no way to determine from this approximation if M is truly hyperbolic and complete. This paper provides a method for proving that a manifold has a complete hyperbolic structure based on the approximations of Snap [7], a program that includes the functionality of SnapPea plus other features. The approximation is done by triangulating M , identifying consistency and completeness equations as described by Neumann and Zagier [13] and Benedetti and Petronio [1] with respect to this triangulation, and then, according to Weeks [21], trying to solve the system of equations using Newton’s Method. This produces an approximate, not actual solution. The method here uses the Kantorovich Theorem [8] to prove that an actual solution exists, thereby assuring that the manifold has a complete hyperbolic structure. Using this, we can definitively prove that every manifold in the SnapPea cusped census has a complete hyperbolic structure.

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].)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.

We derive an explicit formula for the η-invariant of hyperbolic 3-manifolds obtained by Dehn surgeries on some one-cusped hyperbolic 3-manifolds (e.g., hyperbolic knot complements in S3).

authors have recently computed the change in the ^/-invariant under general Riemannian cutting and pasting along closed surfaces.These results are in our preprint, Cutting and pasting and the η-invariant.

Toutes les 3-varietes d'orbites orientables completes ont un volume au moins 0,0000013. Toutes les 3-varietes d'orbites hyperboliques orientables completes cuspides ont un volume au moins (3) 1/2 /24