Volume 8, issue 1 (2008)

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Volume and homology of one-cusped hyperbolic $3$–manifolds

Marc Culler and Peter B Shalen

Algebraic & Geometric Topology 8 (2008) 343–379
Abstract

Let $M$ be a complete, finite-volume, orientable hyperbolic manifold having exactly one cusp. If we assume that ${\pi }_{1}\left(M\right)$ has no subgroup isomorphic to a genus–$2$ surface group and that either (a) ${dim}_{{ℤ}_{p}}{H}_{1}\left(M;{ℤ}_{p}\right)\ge 5$ for some prime $p$, or (b) ${dim}_{{ℤ}_{2}}{H}_{1}\left(M;{ℤ}_{2}\right)\ge 4$, and the subspace of ${H}^{2}\left(M;{ℤ}_{2}\right)$ spanned by the image of the cup product ${H}^{1}\left(M;{ℤ}_{2}\right)×{H}^{1}\left(M;{ℤ}_{2}\right)\to {H}^{2}\left(M;{ℤ}_{2}\right)$ has dimension at most $1$, then $volM>5.06.$ If we assume that ${dim}_{{ℤ}_{2}}{H}_{1}\left(M;{ℤ}_{2}\right)\ge 7$ and that the compact core $N$ of $M$ contains a genus–$2$ closed incompressible surface, then $volM>5.06.$ Furthermore, if we assume only that ${dim}_{{ℤ}_{2}}{H}_{1}\left(M;{ℤ}_{2}\right)\ge 7$, then $volM>3.66.$

Keywords
hyperbolic manifold, cusp, volume, homology, Dehn filling
Primary: 57M50
Secondary: 57M27