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 $2V_{\rm oct}$, where $V_{\rm oct}=3.66\ldots$ denotes the volume of a regular ideal hyperbolic octahedron in $\mathbb{H}^3$. 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 $\mathop{\rm vol} M\le V_{\rm oct}/2$, then $\dim H_1(M;\mathbb{F}_2)\le4$. We also provide new lower bounds for the volumes of closed hyperbolic $3$-manifolds whose cohomology ring over $\mathbb{F}_2$ satisfies certain restrictions; these improve results that were proved in ``Volume and topology$\ldots$.''