  Let M be a complete, finite-volume, orientable hyperbolic manifold having
exactly one cusp. If we assume that pi_1(M) has no subgroup isomorphic to a
genus-2 surface group, and that either (a) H_1(M;Z_p) has dimension at least 5
for some prime p, or (b) H_1(M;Z_2) has dimension at least 4, and the subspace
of H^2(M;Z_2) spanned by the image of the cup product has dimension at most 1,
then vol M > 5.06 If we assume that H_1(M;Z_2) has dimension at least 7, and
that the compact core of M does not contain a genus-2 closed incompressible
surface, then vol M > 5.06.
