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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

DOI: 10.2140/agt.2008.8.343
Abstract

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) dimZpH1(M;Zp)≥ 5 for some prime p, or (b) dimZ2H1(M;Z2)≥ 4, and the subspace of H2(M;Z2) spanned by the image of the cup product H1(M;Z2)× H1(M;Z2)→ H2(M;Z2) has dimension at most 1, then vol M > 5.06. If we assume that dimZ2H1(M;Z2)≥ 7 and that the compact core N of M contains a genus–2 closed incompressible surface, then vol M > 5.06. Furthermore, if we assume only that dimZ2H1(M;Z2)≥ 7, then vol M > 3.66.

Keywords

hyperbolic manifold, cusp, volume, homology, Dehn filling
Mathematical Subject Classification

Primary: 57M50

Secondary: 57M27
References
Publication

Received: 24 August 2007
Revised: 3 February 2007
Accepted: 17 December 2007
Published: 12 May 2008
Authors
Marc Culler
Department of Mathematics (M/C 249)
University of Illinois at Chicago
851 S Morgan St
Chicago, IL 60607-7045
http://www.math.uic.edu/~culler
Peter B Shalen
Department of Mathematics (M/C 249)
University of Illinois at Chicago
851 S Morgan St
Chicago, IL 60607-7045
http://www.math.uic.edu/~shalen