## Algebraic & Geometric Topology

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

#### 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 $volM>5.06.$ If we assume that $dimℤ2H1(M;ℤ2)≥7$ and that the compact core $N$ of $M$ contains a genus–$2$ closed incompressible surface, then $volM>5.06.$ Furthermore, if we assume only that $dimℤ2H1(M;ℤ2)≥7$, then $volM>3.66.$

#### Article information

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

Dates
Revised: 3 February 2007
Accepted: 17 December 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796816

Digital Object Identifier
doi:10.2140/agt.2008.8.343

Mathematical Reviews number (MathSciNet)
MR2443232

Zentralblatt MATH identifier
1160.57012

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

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796816

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