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 $2V_{\mathrm{o}\mathrm{c}\mathrm{t}}$, where $V_{\mathrm{o}\mathrm{c}\mathrm{t}}=3.66\dots$ 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 $M\le V_{\mathrm{o}\mathrm{c}\mathrm{t}}\mathrm{/}2$, then $\dim H_1(M;{\mathbb{F}}_2)\le 4$. 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$\dots$.‘’