The shape of hyperbolic Dehn surgery space.

*(English)*Zbl 1144.57015For a compact orientable \(3\)-manifold \(X\) having boundary a single torus, and whose interior admits a hyperbolic structure of finite volume, a celebrated result of Thurston shows that almost all Dehn fillings produce hyperbolizable closed manifolds. This is seen by proving that the hyperbolic Dehn surgery space \(\mathcal{HDS}(X)\), regarded as a subset of a plane whose points correspond to certain incomplete hyperbolic metrics on the interior of \(X\), contains a neighborhood of infinity. The Dehn fillings correspond to the integer lattice points, so all but finitely many produce hyperbolizable manifolds. Thurston also gave a version when \(X\) has more than one boundary torus.

In this paper, continuing previous work in [Ann. Math. (2) 162, No. 1, 367–421 (2005; Zbl 1087.57011)] and other articles, the authors obtain more precise information about \(\mathcal{HDS}(X)\). Writing \(T\) for the boundary torus, \(\mathcal{HDS}(X)\) may be regarded in a natural way as a subset of \(H_1(T;\mathbb{R})\), The first main result states that \(\mathcal{HDS}(X)\) contains the complement of a disk of radius \(7.5832\), centered at the origin. The second is a version of this when \(X\) has more than one boundary torus.

The proofs of these results use deep analytic techniques, which yield a great deal of additional information. For example, for Dehn fillings outside the disk of radius \(7.5832\), the difference in volume between the complete structure on \(X\) and the volume of the filled manifold is at most \(0.198\). More technical results give strong control on the geometry of the manifolds under deformations.

A key technical idea is to work not with the cone manifolds that result from completing the metrics, but with the compact hyperbolic manifolds with boundary that result from truncating the cusps along flat tori. The infinitesimal deformations of such manifolds can be viewed as cohomology classes, and under certain conditions harmonic representatives can be found. One consequence of this is a local rigidity result: under certain hypotheses on the geometry of the boundary tori, there are no infinitesimal deformations keeping the Dehn surgery coefficients constant. This implies that the Dehn surgery coefficients give a local parameterization for cone manifolds. Once the local rigidity and local parameterization theorems are established, the uniform bounds can be deduced using the general approach of the authors’ article mentioned above. The use of hyperbolic manifolds with boundary, as opposed to cone manifolds, produces subtly different estimates in the case of multiple boundary tori.

Unlike the Hodge theory developed in the authors’ previous work, the new version accommodates cone angles greater than \(2\pi\). The resulting harmonic deformation theory has been used by K. Bromberg [J. Am. Math. Soc. 17, No. 4, 783–826 (2004; Zbl 1061.30037), Ann. Math. (2) 166, No. 1, 77–93 (2007; Zbl 1137.30014)] in his proof of the Bers Density Conjecture, as well as by K. Bromberg and J. Brock to obtain more general versions of the Density Conjecture [Acta Math. 192, No. 1, 33–93 (2004; Zbl 1055.57020)].

In this paper, continuing previous work in [Ann. Math. (2) 162, No. 1, 367–421 (2005; Zbl 1087.57011)] and other articles, the authors obtain more precise information about \(\mathcal{HDS}(X)\). Writing \(T\) for the boundary torus, \(\mathcal{HDS}(X)\) may be regarded in a natural way as a subset of \(H_1(T;\mathbb{R})\), The first main result states that \(\mathcal{HDS}(X)\) contains the complement of a disk of radius \(7.5832\), centered at the origin. The second is a version of this when \(X\) has more than one boundary torus.

The proofs of these results use deep analytic techniques, which yield a great deal of additional information. For example, for Dehn fillings outside the disk of radius \(7.5832\), the difference in volume between the complete structure on \(X\) and the volume of the filled manifold is at most \(0.198\). More technical results give strong control on the geometry of the manifolds under deformations.

A key technical idea is to work not with the cone manifolds that result from completing the metrics, but with the compact hyperbolic manifolds with boundary that result from truncating the cusps along flat tori. The infinitesimal deformations of such manifolds can be viewed as cohomology classes, and under certain conditions harmonic representatives can be found. One consequence of this is a local rigidity result: under certain hypotheses on the geometry of the boundary tori, there are no infinitesimal deformations keeping the Dehn surgery coefficients constant. This implies that the Dehn surgery coefficients give a local parameterization for cone manifolds. Once the local rigidity and local parameterization theorems are established, the uniform bounds can be deduced using the general approach of the authors’ article mentioned above. The use of hyperbolic manifolds with boundary, as opposed to cone manifolds, produces subtly different estimates in the case of multiple boundary tori.

Unlike the Hodge theory developed in the authors’ previous work, the new version accommodates cone angles greater than \(2\pi\). The resulting harmonic deformation theory has been used by K. Bromberg [J. Am. Math. Soc. 17, No. 4, 783–826 (2004; Zbl 1061.30037), Ann. Math. (2) 166, No. 1, 77–93 (2007; Zbl 1137.30014)] in his proof of the Bers Density Conjecture, as well as by K. Bromberg and J. Brock to obtain more general versions of the Density Conjecture [Acta Math. 192, No. 1, 33–93 (2004; Zbl 1055.57020)].

Reviewer: Darryl McCullough (Norman)

##### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

57N16 | Geometric structures on manifolds of high or arbitrary dimension |

##### Keywords:

3-manifold; hyperbolic; cone manifold; filling; Dehn filling; volume; deformation; infinitesimal; surgery space; Weitzenbock; harmonic
PDF
BibTeX
XML
Cite

\textit{C. D. Hodgson} and \textit{S. P. Kerckhoff}, Geom. Topol. 12, No. 2, 1033--1090 (2008; Zbl 1144.57015)

**OpenURL**

##### References:

[1] | M Atiyah, Elliptic boundary value problems, Annals of Math. Studies 57, Princeton University Press (1965) |

[2] | S Boyer, X Zhang, On Culler-Shalen seminorms and Dehn filling, Ann. of Math. \((2)\) 148 (1998) 737 · Zbl 1007.57016 |

[3] | J F Brock, K W Bromberg, On the density of geometrically finite Kleinian groups, Acta Math. 192 (2004) 33 · Zbl 1055.57020 |

[4] | K Bromberg, Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives, J. Amer. Math. Soc. 17 (2004) 783 · Zbl 1061.30037 |

[5] | K Bromberg, Projective structures with degenerate holonomy and the Bers density conjecture, Ann. of Math. \((2)\) 166 (2007) 77 · Zbl 1137.30014 |

[6] | D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of \(3\)-manifolds, Invent. Math. 118 (1994) 47 · Zbl 0842.57013 |

[7] | M Culler, P B Shalen, Varieties of group representations and splittings of \(3\)-manifolds, Ann. of Math. \((2)\) 117 (1983) 109 · Zbl 0529.57005 |

[8] | C D Hodgson, Degeneration and regeneration of geometric structures on \(3\)-manifolds, PhD thesis, Princeton University (1986) |

[9] | C D Hodgson, S P Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998) 1 · Zbl 0919.57009 |

[10] | C D Hodgson, S P Kerckhoff, Harmonic deformations of hyperbolic 3-manifolds, London Math. Soc. Lecture Note Ser. 299, Cambridge Univ. Press (2003) 41 · Zbl 1051.57018 |

[11] | C D Hodgson, S P Kerckhoff, Universal bounds for hyperbolic Dehn surgery, Ann. of Math. \((2)\) 162 (2005) 367 · Zbl 1087.57011 |

[12] | L HĂ¶rmander, Linear partial differential operators, Springer Verlag (1976) · Zbl 0321.35001 |

[13] | W D Neumann, D Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307 · Zbl 0589.57015 |

[14] | J Purcell, Cusp shapes under cone deformation · Zbl 1182.57015 |

[15] | M Spivak, A comprehensive introduction to differential geometry. Vol. III, Publish or Perish (1979) · Zbl 0439.53001 |

[16] | W Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.