The minimal volume orientable hyperbolic 2-cusped 3-manifolds

Abstract

Abstract. We prove that the Whitehead link complement and the (−2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 × Catalan's constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.

Closed surface bundles of least volume

“…Proof of Theorem 4.2 Suppose M has at least two cusps; then the conclusion is exactly Theorem 3.6 of [1]. So suppose M has one cusp.…”

“…The first, Theorem 4.3, is the qualitative statement that the minimal volume † g -bundles, for large g , must be Dehn fillings on the Whitehead sibling W . This is proved by combining a soft geometric limit argument with work of Agol [1] on volumes of cusped manifolds. Then, in Theorem 5.1, we sift through the large number of † g -bundles that arise from W and find the one with least volume.…”

“…2; 3; 8/-pretzel link shown in Figure 3, and is manifold m125 in the SnapPea census [2]. Like the Whitehead complement, it can be built by gluing the sides of a regular ideal octahedron in H 3 (see [1]); thus its volume is…”

Some Hyperbolic Space Forms With Few Generated Fundamental Groups

“…This manifold is obtained by identifying the faces of a single ideal tetrahedron; it has the first integral homology group and the isometry group both isomorphic to Z 2 . Now applying Agol's result [2], we have: Finally, we observe that the one-relator group G in Theorem 4.1 is properly 3-realizable in the sense of [5], i.e., there exists a compact 2-polyhedron K with π 1 (K) ∼ = G whose universal cover K is proper homotopy equivalent to a 3-manifold. In our case, K is a spine of the space form M (for example, the spine which corresponds to the finite presentation of G) and K is proper homotopy equivalent to H 3 .…”

Small volume link orbifolds

“…For, any multi-cusped manifold of volume ≤ 2.848 would have an infinite sequence of 1-cusped Dehn fillings also satisfying this volume bound, which would contradict their enumeration. In fact, Agol showed [2] the smallest volume multi-cusped manifold has Vol(N ) = 3.6638 . .…”