##
**Finite Dehn surgery on knots.**
*(English)*
Zbl 0936.57010

This paper studies the problem which Dehn surgery on a knot in a closed orientable 3-manifold can produce 3-manifolds wilh finite [resp. cyclic] fundamental group. Such a surgery is called a finite [resp. cyclic] surgery and its surgery slope is called a finite [resp. cyclic] surgery slope. It is shown that if the exterior \(M\) of a knot \(K\) is hyperbolic, then (1) there are at most six finite/cyclic surgeries on \(K\), and the distance between two finite/cyclic surgery slopes of \(K\) is at most 5; (2) the distance between a finite/cyclic surgery slope and a cyclic surgery slope of \(K\) is at most 2. The result (1) sharply improves the bounds obtained by S. A. Bleiler and C. D. Hodgson [Topology 35, No. 3, 809-833 (1996; Zbl 0863.57009)] by using the \(2\pi\) theorem of Gromov and Thurston. Moreover, the authors conjecture that the number 5 in (1) is replaced with 3. Further results of the case when \(K\) is a knot in \(S^3\) are also obtained.

The approach in this paper follows that developed by M. Culler, C. McA. Gordon, J. Luecke and P. B. Shalen [Ann. Math., II. Ser. 125, 237-300 (1987; Zbl 0633.57006)] for analyzing cyclic surgery slopes, and sharp bounds on the Culler-Shalen norm of finite surgery slopes are obtained.

Studies of the case when \(M\) contains an essential torus are also contained. In particular, it is shown that a satellite knot \(K\) in \(S^3\) admits a non-trivial finite cyclic surgery, then \(K\) is a cabled knot over a torus knot: this gives the complete classification of finite surgeries on satellite knots in \(S^3\). The paper also fcontains various examples which show the extent to which the results are sharp.

The approach in this paper follows that developed by M. Culler, C. McA. Gordon, J. Luecke and P. B. Shalen [Ann. Math., II. Ser. 125, 237-300 (1987; Zbl 0633.57006)] for analyzing cyclic surgery slopes, and sharp bounds on the Culler-Shalen norm of finite surgery slopes are obtained.

Studies of the case when \(M\) contains an essential torus are also contained. In particular, it is shown that a satellite knot \(K\) in \(S^3\) admits a non-trivial finite cyclic surgery, then \(K\) is a cabled knot over a torus knot: this gives the complete classification of finite surgeries on satellite knots in \(S^3\). The paper also fcontains various examples which show the extent to which the results are sharp.

Reviewer: M.Sakuma (Osaka)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57R65 | Surgery and handlebodies |

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

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

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\textit{S. Boyer} and \textit{X. Zhang}, J. Am. Math. Soc. 9, No. 4, 1005--1050 (1996; Zbl 0936.57010)

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### References:

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