Universal bounds for hyperbolic Dehn surgery.

*(English)*Zbl 1087.57011Let \(X\) be a noncompact, finite volume, orientable hyperbolic \(3\)-manifold. Then it is the interior of a compact \(3\)-manifold with a finite number of torus boundary components. Thurston’s hyperbolic Dehn surgery theorem states that, for all but a finite number of Dehn surgeries on each boundary component, the resulting closed \(3\)-manifold has a hyperbolic structure. The paper under review gives the first universal bound on the number of nonhyperbolic Dehn surgeries per boundary torus, independent of the manifold \(X\). If there is only one cusp, there are at most \(60\) nonhyperbolic Dehn surgeries; if there are multiple cusps, at most \(114\) surgery curves must be excluded from each boundary torus.

There is another approach to such a universal bound. M. Lackenby [Invent. Math. 140, No. 2, 243–282 (2000; Zbl 0947.57016)] and I. Agol [Geom. Topol. 4, 431–449 (2000; Zbl 0959.57009)] independently showed that if the flat geodesic length of a slope \(\gamma\) on a boundary torus is greater than \(6\), then the Dehn filled manifold is irreducible with infinite word hyperbolic fundamental group. Then I. Agol [loc. cit.] proved that, when there is a single cusp, at most \(12\) surgeries fail to be irreducible or infinite word hyperbolic, by using C. Cao and G. R. Meyerhoff’s result [Invent. Math. 146, No. 3, 451–478 (2001; Zbl 1028.57010)]. This is remarkably close to the largest known number \(10\) of nonhyperbolic Dehn surgeries.

For simplicity, assume that \(X\) has a single cusp. The normalized length of a curve \(\gamma\) on the boundary torus is defined to be the geodesic length after rescaling the metric on the torus to have area \(1\). The main result shows that if the normalized length of \(\gamma\) is at least \(7.515\), then it is possible to deform the complete hyperbolic structure through cone-manifold structures on \(X(\gamma)\) with \(\gamma\) bounding a singular meridian disk until the cone angle reaches \(2\pi\). This gives a smooth hyperbolic structure on \(X(\gamma)\).

The proof has two parts. One part is analytic, showing that under the normalized length hypothesis on \(\gamma\), there is a lower bound to the tube radius for any of cone-manifold structures on \(X(\gamma)\) with angle at most \(2\pi\), based on the local rigidity theory for cone-manifolds. The second part shows that, under certain geometric conditions, no degeneration of the hyperbolic structure is possible.

In the final section, the authors prove two more results. It is natural to ask whether every closed hyperbolic \(3\)-manifold can be obtained by starting with a noncompact finite volume \(3\)-manifold with one cusp and increasing the cone angle from \(0\) to \(2\pi\). In fact, they prove that if a closed hyperbolic \(3\)-manifold \(M\) has a simple closed geodesic \(\tau\) in \(M\) with length at most \(0.111\), then the hyperbolic structure on \(M\) can be deformed to a complete hyperbolic structure on \(M-\tau\) by decreasing the cone angle along \(\tau\) from \(2\pi\) to \(0\). Secondly, let \(X\) be a complete finite volume hyperbolic \(3\)-manifold with one cusp and let \(\gamma\) be a surgery curve with normalized length at least \(7.515\). Then they show that the difference between the volume of \(X(\gamma)\) and \(X\) is at most \(0.329\), and also that every closed hyperbolic \(3\)-manifold with a closed geodesic of length at most \(0.162\) has volume at least \(1.701\).

There is another approach to such a universal bound. M. Lackenby [Invent. Math. 140, No. 2, 243–282 (2000; Zbl 0947.57016)] and I. Agol [Geom. Topol. 4, 431–449 (2000; Zbl 0959.57009)] independently showed that if the flat geodesic length of a slope \(\gamma\) on a boundary torus is greater than \(6\), then the Dehn filled manifold is irreducible with infinite word hyperbolic fundamental group. Then I. Agol [loc. cit.] proved that, when there is a single cusp, at most \(12\) surgeries fail to be irreducible or infinite word hyperbolic, by using C. Cao and G. R. Meyerhoff’s result [Invent. Math. 146, No. 3, 451–478 (2001; Zbl 1028.57010)]. This is remarkably close to the largest known number \(10\) of nonhyperbolic Dehn surgeries.

For simplicity, assume that \(X\) has a single cusp. The normalized length of a curve \(\gamma\) on the boundary torus is defined to be the geodesic length after rescaling the metric on the torus to have area \(1\). The main result shows that if the normalized length of \(\gamma\) is at least \(7.515\), then it is possible to deform the complete hyperbolic structure through cone-manifold structures on \(X(\gamma)\) with \(\gamma\) bounding a singular meridian disk until the cone angle reaches \(2\pi\). This gives a smooth hyperbolic structure on \(X(\gamma)\).

The proof has two parts. One part is analytic, showing that under the normalized length hypothesis on \(\gamma\), there is a lower bound to the tube radius for any of cone-manifold structures on \(X(\gamma)\) with angle at most \(2\pi\), based on the local rigidity theory for cone-manifolds. The second part shows that, under certain geometric conditions, no degeneration of the hyperbolic structure is possible.

In the final section, the authors prove two more results. It is natural to ask whether every closed hyperbolic \(3\)-manifold can be obtained by starting with a noncompact finite volume \(3\)-manifold with one cusp and increasing the cone angle from \(0\) to \(2\pi\). In fact, they prove that if a closed hyperbolic \(3\)-manifold \(M\) has a simple closed geodesic \(\tau\) in \(M\) with length at most \(0.111\), then the hyperbolic structure on \(M\) can be deformed to a complete hyperbolic structure on \(M-\tau\) by decreasing the cone angle along \(\tau\) from \(2\pi\) to \(0\). Secondly, let \(X\) be a complete finite volume hyperbolic \(3\)-manifold with one cusp and let \(\gamma\) be a surgery curve with normalized length at least \(7.515\). Then they show that the difference between the volume of \(X(\gamma)\) and \(X\) is at most \(0.329\), and also that every closed hyperbolic \(3\)-manifold with a closed geodesic of length at most \(0.162\) has volume at least \(1.701\).

Reviewer: Masakazu Teragaito (Hiroshima)

##### MSC:

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

57N10 | Topology of general \(3\)-manifolds (MSC2010) |