Volume and homology of one-cusped hyperbolic 3–manifolds

Culler, Marc; Shalen, Peter B

doi:10.2140/agt.2008.8.343

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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 $π_{1} (M)$ has no subgroup isomorphic to a genus– $2$ surface group and that either (a) ${dim}_{ℤ_{p}} H_{1} (M; ℤ_{p}) \geq 5$ for some prime $p$ , or (b) ${dim}_{ℤ_{2}} H_{1} (M; ℤ_{2}) \geq 4$ , and the subspace of $H^{2} (M; ℤ_{2})$ spanned by the image of the cup product $H^{1} (M; ℤ_{2}) \times H^{1} (M; ℤ_{2}) \to H^{2} (M; ℤ_{2})$ has dimension at most $1$ , then $vol M > 5.06 .$ If we assume that ${dim}_{ℤ_{2}} H_{1} (M; ℤ_{2}) \geq 7$ 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 ${dim}_{ℤ_{2}} H_{1} (M; ℤ_{2}) \geq 7$ , then $vol M > 3.66 .$

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

Volume 8, issue 1 (2008)