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

Marc Culler; Peter B Shalen

doi:10.2140/agt.2008.8.343

Algebraic & Geometric Topology

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

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

Marc Culler and Peter B Shalen

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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 $vol M > 3.66 .$

Article information

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

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

Permanent link to this document
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

Keywords
hyperbolic manifold cusp volume homology Dehn filling

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

References

I Agol, Tameness of hyperbolic $3$–manifolds
arXiv: math.GT/0405568
I Agol, M Culler, P B Shalen, Singular surfaces, mod 2 homology, and hyperbolic volume, I., to appear in Trans. Amer. Math. Soc.
arXiv: math.GT/0506396
Zentralblatt MATH: 1195.57037
Digital Object Identifier: doi:10.1090/S0002-9947-10-04362-X
I Agol, M Culler, P B Shalen, Dehn surgery, homology and hyperbolic volume, Algebr. Geom. Topol. 6 (2006) 2297–2312
Mathematical Reviews (MathSciNet): MR2286027
Zentralblatt MATH: 1129.57019
Digital Object Identifier: doi:10.2140/agt.2006.6.2297
J W Anderson, R D Canary, M Culler, P B Shalen, Free Kleinian groups and volumes of hyperbolic $3$–manifolds, J. Differential Geom. 43 (1996) 738–782
Mathematical Reviews (MathSciNet): MR1412683
Digital Object Identifier: doi:10.4310/jdg/1214458530
Project Euclid: euclid.jdg/1214458530
M D Baker, On boundary slopes of immersed incompressible surfaces, Ann. Inst. Fourier $($Grenoble$)$ 46 (1996) 1443–1449
Mathematical Reviews (MathSciNet): MR1427132
Zentralblatt MATH: 0864.57015
Digital Object Identifier: doi:10.5802/aif.1555
R Benedetti, C Petronio, Lectures on hyperbolic geometry, Universitext, Springer, Berlin (1992)
Mathematical Reviews (MathSciNet): MR1219310
Zentralblatt MATH: 0768.51018
K B ör öczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978) 243–261
Mathematical Reviews (MathSciNet): MR512399
Digital Object Identifier: doi:10.1007/BF01902361
K B ör öczky, A Florian, Über die dichteste Kugelpackung im hyperbolischen Raum, Acta Math. Acad. Sci. Hungar 15 (1964) 237–245
Mathematical Reviews (MathSciNet): MR0160155
Zentralblatt MATH: 0125.39803
Digital Object Identifier: doi:10.1007/BF01897041
D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 19 (2006) 385–446
Mathematical Reviews (MathSciNet): MR2188131
Zentralblatt MATH: 1090.57010
Digital Object Identifier: doi:10.1090/S0894-0347-05-00513-8
M Culler, J DeBlois, P B Shalen, Incompressible surfaces, hyperbolic volume, Heegaard genus and homology, in preparation
Zentralblatt MATH: 1187.57020
Digital Object Identifier: doi:10.4310/CAG.2009.v17.n2.a1
M Culler, P B Shalen, Paradoxical decompositions, $2$–generator Kleinian groups, and volumes of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 5 (1992) 231–288
Mathematical Reviews (MathSciNet): MR1135928
J Hass, S Wang, Q Zhou, On finiteness of the number of boundary slopes of immersed surfaces in $3$–manifolds, Proc. Amer. Math. Soc. 130 (2002) 1851–1857
Mathematical Reviews (MathSciNet): MR1887034
Digital Object Identifier: doi:10.1090/S0002-9939-01-06262-1
A E Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982) 373–377
Mathematical Reviews (MathSciNet): MR658066
Zentralblatt MATH: 0502.57005
Digital Object Identifier: doi:10.2140/pjm.1982.99.373
Project Euclid: euclid.pjm/1102734021
J Hempel, $3$–manifolds, AMS Chelsea Publishing, Providence, RI (2004) Reprint of the 1976 original
Mathematical Reviews (MathSciNet): MR2098385
T Jørgensen, A Marden, Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990) 47–72
Mathematical Reviews (MathSciNet): MR1060898
Zentralblatt MATH: 0738.30032
Digital Object Identifier: doi:10.7146/math.scand.a-12292
A G Kurosh, The theory of groups, Chelsea Publishing Co., New York (1960) (2 volumes) Translated from the Russian and edited by K A Hirsch
Mathematical Reviews (MathSciNet): MR0109842
U Oertel, Incompressible maps of surfaces and Dehn filling, Topology Appl. 136 (2004) 189–204
Mathematical Reviews (MathSciNet): MR2023417
Zentralblatt MATH: 1039.57011
Digital Object Identifier: doi:10.1016/S0166-8641(03)00219-0
C Petronio, J Porti, Negatively oriented ideal triangulations and a proof of Thurston's hyperbolic Dehn filling theorem, Expo. Math. 18 (2000) 1–35
Mathematical Reviews (MathSciNet): MR1751141
Zentralblatt MATH: 0977.57011
G P Scott, Compact submanifolds of $3$–manifolds, J. London Math. Soc. $(2)$ 7 (1973) 246–250
Mathematical Reviews (MathSciNet): MR0326737
Zentralblatt MATH: 0266.57001
Digital Object Identifier: doi:10.1112/jlms/s2-7.2.246
P B Shalen, P Wagreich, Growth rates, $Z\sb p$–homology, and volumes of hyperbolic $3$–manifolds, Trans. Amer. Math. Soc. 331 (1992) 895–917
Mathematical Reviews (MathSciNet): MR1156298
J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170–181
Mathematical Reviews (MathSciNet): MR0175956
Zentralblatt MATH: 0135.05201
Digital Object Identifier: doi:10.1016/0021-8693(65)90017-7
J Weeks, C Hodgson, Census of closed hyperbolic $3$–manifolds Available at \setbox0\makeatletter\@url http://www.geometrygames.org/SnapPea/ {\unhbox0
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