Volume and topology of bounded and closed hyperbolic 3-manifolds, II

Let N be a compact, orientable hyperbolic 3-manifold whose boundary is a connected totally geodesic surface of genus 2. If N has Heegaard genus at least 5, then its volume is greater than 2V ₎₂ₓ, where V ₎₂ₓ=3. 66 denotes the volume of a regular ideal hyperbolic octahedron in H³. This improves the lower bound given in our earlier paper ``Volume and topology of bounded and closed hyperbolic 3-manifolds. '' One ingredient in the improved bound is that in a crucial case, instead of using a single ``muffin'' in N in the sense of Kojima and Miyamoto, we use two disjoint muffins. By combining the result about manifolds with geodesic boundary with the (2k-1) theorem and results due to Agol-Culler-Shalen and Shalen-Wagreich, we show that if M is a closed, orientable hyperbolic 3-manifold with vol M V ₎₂ₓ/2, then H₁ (M;F₂) 4. We also provide new lower bounds for the volumes of closed hyperbolic 3-manifolds whose cohomology ring over F₂ satisfies certain restrictions; these improve results that were proved in ``Volume and topology. ''