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 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.

Keywords
hyperbolic manifold, cusp, volume, homology, Dehn filling
Mathematical Subject Classification 2000
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