Volumes of hyperbolic Haken manifolds, I

Abstract

In [14] a program was initiated for using the topological theory of 3-manifolds to obtain lower bounds for volumes of hyperbolic 3-manifolds. In [1], by a combination of new geometric ideas with relatively standard (but specifically 3-dimensional) topological techniques, we showed that every closed, orientable hyperbolic 3-manifold whose first Betti number is at least 3 has volume exceeding 0.92. By contrast, the best known lower bound [10,5] for the volume of an arbitrary closed hyperbolic 3-manifold is approximately 0.0012. In [3] we showed that every closed, orientable hyperbolic 3-manifold whose first Betti number is 2 has volume exceeding 0.34. The proof depended on supplementing the results and techniques of [1] with ingenious elementary arguments due to Zagier [11] and numerical computations. In the present paper we shall show that if one excludes certain special manifolds, such as fiber bundles over S 1, then the lower bound of 0.34 also holds for hyperbolic 3-manifolds with Betti number 1. The proof depends heavily on the results of [1] and [3], but it involves much deeper topological ideas than these papers. The new topological results needed for the proof occupy most of the present paper. To some extent these results have the flavor of general topology, but the proofs make use of such specifically low-dimensional techniques as the characteristic submanifold theory [9, 8], the interaction between trees and incompressible surfaces, and Scott's theorem [12] that surface groups are locally extended residually finite. Before giving a precise statement of our main result we must review a few elementary notions from 3-manifold theory.