Algebraic & Geometric Topology

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

Marc Culler and Peter B Shalen

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Let M be a complete, finite-volume, orientable hyperbolic manifold having exactly one cusp. If we assume that π1(M) has no subgroup isomorphic to a genus–2 surface group and that either (a) dimpH1(M;p)5 for some prime p, or (b) dim2H1(M;2)4, and the subspace of H2(M;2) spanned by the image of the cup product H1(M;2)×H1(M;2)H2(M;2) has dimension at most 1, then volM>5.06. If we assume that dim2H1(M;2)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 dim2H1(M;2)7, then volM>3.66.

Article information

Algebr. Geom. Topol., Volume 8, Number 1 (2008), 343-379.

Received: 24 August 2007
Revised: 3 February 2007
Accepted: 17 December 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57M27: Invariants of knots and 3-manifolds

hyperbolic manifold cusp volume homology Dehn filling


Culler, Marc; Shalen, Peter B. Volume and homology of one-cusped hyperbolic $3$–manifolds. Algebr. Geom. Topol. 8 (2008), no. 1, 343--379. doi:10.2140/agt.2008.8.343.

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