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 2Voct, where Voct=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 log(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≤ Voct/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….''