Volume and Homology for Hyperbolic 3-Orbifolds

Let $\mathfrak{M}$ be a closed, orientable, hyperbolic 3-orbifold such that ${\pi }_1(\mathfrak{M})$ contains no hyperbolic triangle group. We show that strict upper bounds of 0.07625, 0.1525 and 0.22875 for ${\rm vol}\ {\mathfrak M}$implyrespectiveupperboundsof23,43and79for$\dim H_1({\mathfrak M};{\mathbb F}_2 )$\mathrm{.}Strongerresultsholdifweassumethatthesingularset$\Sigma$ is a link; specifically, under this assumption, strict upper bounds of 0.305, 0.4575, 0.61, 0.7625 and 0.915 for $\mathrm{v}\mathrm{o}\mathrm{l}\mathfrak{M}$ imply respective upper bounds of 7, 13, 14, 28 and 29 for ${\rm dim}\ H_1({\mathfrak M};{\mathbb F}_2 )$\mathrm{.}Irreducibilityassumptionsontheunderlyingmanifold$|{\mathfrak M}|$of${\mathfrak M}$, and of the underlying manifolds of certain coverings of $\mathfrak{M}$, also give stronger results. The upper bounds on $\dim H_1(\mathfrak{M};{\mathbb{F}}_2)$ for an orbifold ${\mathfrak M}$ whose volume is subject to a suitable upper bound are deduced from upper bounds on $\mathrm{d}\mathrm{i}\mathrm{m}{H}_1(\mathrm{\mid }\mathfrak{M}\mathrm{\mid };{\mathbb{F}}_2)$ for an orbifold $\mathfrak{M}$ whose volume is subject to a suitable upper bound.