Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Volume and Homology for Hyperbolic $3$-Orbifolds, I   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 $\widetilde{\mathfrak M}$ of ${\mathfrak M}$, and if ${\rm
vol} {\mathfrak M}\le1.72$, then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 15$.
Furthermore, if ${\rm vol} {\mathfrak M}\le1.22$ then $\dim H_1({\mathfrak
M};{\mathbb Z}_2)\le 11$, and if ${\rm vol} {\mathfrak M}\le0.61$ 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
$|\widetilde{\mathfrak M}|$.

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The categories of this paper are: nucl-ex
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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ph
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The categories of this paper are: q-alg