Lower Bounds on Volumes of Hyperbolic Haken 3-manifolds

In this paper, we find lower bounds for the volumes of certain hyperbolic Haken 3manifolds. The theory of Jorgensen and Thurston shows that the volumes of hyperbolic 3-manifolds are well-ordered, but no one has been able to find the smallest one. The best known result for closed manifolds is that the smallest closed hyperbolic 3-manifold has volume > 0.16668, proven by Gabai, Meyerhoff, and Thurston [11]. Their proof involves extensive rigorous computations. The smallest known closed 3-manifold is the Weeks manifold W , V ol(W ) ≈ 0.94270 [26], which is the 2-fold branched cover over the knot 949 [24]. One could also ask for the smallest hyperbolic manifolds with certain characteristics. The smallest non-compact, disoriented hyperbolic 3-manifold is the Gieseking manifold, which is double covered by the figure eight knot complement, proven by Colin Adams [1]. Its volume = V3 ≈ 1.01494, since it is obtained by pairwise gluing the faces of the regular ideal tetrahedron in H, which has this volume. A recent result of Cao and Meyerhoff [4] shows that the smallest oriented noncompact hyperbolic 3-manifolds are the figure-eight knot complement and its sister manifold, which have volume 2V3 ≈ 2.03. Kojima and Miyamoto [16, 17] have found the smallest hyperbolic 3-manifolds with totally geodesic boundary, which include Thurston’s tripos manifold [22]. Culler and Shalen have a series of papers deriving lower bounds for volumes of closed hyperbolic 3-manifolds M, where dim(H1(M ;Q)) = β1(M) ≥ 1. They found V ol(M) > .34, where one excludes “fibroids” if β1(M) = 1 [7, 8]. Culler, Shalen, and Hersonsky showed that if β1(M) ≥ 3, then V ol(M) ≥ .94689 > V ol(W ), which shows that the smallest volume 3-manifold must have β1(M) ≤ 2 [6]. Joel Hass has shown that there is an upper bound to the genus of an acylindrical surface in terms of the volume of a manifold [13]. One consequence of our results is that if β1(M) = 2 or β1(M) = 1 and M is not fibred over S , then V ol(M) ≥ 4 5 V3. If M contains an acylindrical surface S, then V ol(M) ≥ −2V3χ(S). Section 2 gives the necessary definitions and the statement of the main theorem. Section 3 gives some examples. Sections 4-6 are devoted to the proof of the main theorem. Sections 7-9 give applications of the main theorem. Acknowledgements: We thank Daryl Cooper, Mike Freedman, Darren Long, Pat Shanahan, and Bill Thurston for helpful conversations.