Volume and Homology for Hyperbolic $3$-Orbifolds, I

Abstract

Let $\mathfrak{M}$ be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that ${\pi }_1(\mathfrak{M})$ contains no hyperbolic triangle group. We show that if the underlying manifold $|\mathfrak{M}|$ is irreducible, and $|\mathfrak{M}|$ is irreducible for every two-sheeted (orbifold) covering $\tilde{\mathfrak{M}}$ of $\mathfrak{M}$, and if $\mathrm{vol}\mathfrak{M}\le 1.72$, then $\dim H_1(\mathfrak{M};{\mathbb{Z}}_2)\le 15$. Furthermore, if $\mathrm{vol}\mathfrak{M}\le 1.22$ then $\dim H_1(\mathfrak{M};{\mathbb{Z}}_2)\le 11$, and if $\mathrm{vol}\mathfrak{M}\le 0.61$ then $\dim H_1(\mathfrak{M};{\mathbb{Z}}_2)\le 7$. The proof is an application of results that will be used in the sequel to this paper to obtain qualitatively similar results without the assumption of irreducibility of $|\mathfrak{M}|$ and $|\tilde{\mathfrak{M}}|$.

99 pages

References in corpus (1)

• The minimal volume orientable hyperbolic 2-cusped 3-manifolds