The diameter of the set of boundary slopes of a knot

Let K be a tame knot with irreducible exterior M(K) in a closed, connected, orientable 3--manifold Sigma such that pi_1(Sigma) is cyclic. If infinity is not a strict boundary slope, then the diameter of the set of strict boundary slopes of K, denoted d_K, is a numerical invariant of K. We show that either (i) d_K >= 2 or (ii) K is a generalized iterated torus knot. The proof combines results from Culler and Shalen [Comment. Math. Helv. 74 (1999) 530-547] with a result about the effect of cabling on boundary slopes.Comment: This is the version published by Algebraic & Geometric Topology on 29 August 200

Singular surfaces, mod 2 homology, and hyperbolic volume, II

If M is a closed simple 3-manifold whose fundamental group contains a genus-g surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least max(3g-1,6), we show that M contains a closed, incompressible surface of genus at most g. This improves the main topological result of part I, in which the the same conclusion was obtained under the stronger hypothesis that the dimension of H_1(M;Z_2) is at least 4g-1. As an application we show that if M is a closed orientable hyperbolic 3-manifold with volume at most 3.08, then H_1(M;Z_2) has dimension at most 5.Comment: 23 pages. This version incorporates suggestions from the referee and adds a new section giving examples showing that the main theorem is almost sharp for genus 2. The examples have mod 2 homology of rank 4 and their fundamental groups contain genus 2 surface groups, but they have no closed incompressible surface

Margulis numbers for Haken manifolds

For every closed hyperbolic Haken 3-manifold and, more generally, for any hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If M has non-zero first Betti number, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.Comment: 25 pages. Some statements were clarified some typos were corrected and some of the propositions were generalize

Dehn surgery, homology and hyperbolic volume

If a closed, orientable hyperbolic 3--manifold M has volume at most 1.22 then H_1(M;Z_p) has dimension at most 2 for every prime p not 2 or 7, and H_1(M;Z_2) and H_1(M;Z_7) have dimension at most 3. The proof combines several deep results about hyperbolic 3--manifolds. The strategy is to compare the volume of a tube about a shortest closed geodesic C in M with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from M by Dehn surgeries on C.Comment: This is the version published by Algebraic & Geometric Topology on 8 December 200