  Let N be a compact, orientable hyperbolic 3-manifold with connected, totally
geodesic boundary of genus 2. If N has Heegaard genus at least 5, then its
volume is greater than 6.89. The proof of this result uses the following
dichotomy: either N has a long return path (defined by Kojima-Miyamoto), or N
has an embedded, codimension-0 submanifold X with incompressible boundary $T
\sqcup \partial N$, where T is the frontier of X in N, which is not a book of
I-bundles. As an application of this result, we show that if M is a closed,
orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 5,
and if the image in H^2(M;Z_2) of the cup product map has image of dimension at
most 1, then M has volume greater than 3.44.
