A small arithmetic hyperbolic three-manifold.

*(English)*Zbl 0621.57006It was shown by Jørgenson and Thurston that there is a minimal element \(v_ 1\) in the set of volumes of complete orientable hyperbolic three- manifolds. Let M be the complete orientable hyperbolic 3-manifold resulting from (5,1) Dehn surgery on the complement of the figure-eight knot K in \(S^ 3\). It is established in this paper that M is arithmetic. On the other hand, the author and Jørgenson have shown that there is an arithmetic manifold M’ with vol M’\(<vol M\).

Reviewer: G.Soifer

##### MSC:

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

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

##### Keywords:

volumes of complete orientable hyperbolic three-manifolds; (5,1) Dehn surgery on the complement of the figure-eight knot; arithmetic manifold
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##### References:

[1] | A. Borel, Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 1 – 33. · Zbl 0473.57003 |

[2] | Ted Chinburg and Eduardo Friedman, The smallest arithmetic hyperbolic three-orbifold, Invent. Math. 86 (1986), no. 3, 507 – 527. · Zbl 0643.57011 |

[3] | H. J. Godwin, On quartic fields of signature one with small discriminant, Quart. J. Math. Oxford Ser. (2) 8 (1957), 214 – 222. · Zbl 0079.05704 |

[4] | R. Meyerhoff, A lower bound for the volume of hyperbolic \( 3\)-manifolds, preprint (1982). |

[5] | John Milnor, Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 9 – 24. · Zbl 0486.01006 |

[6] | Robert Riley, A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77 (1975), 281 – 288. · Zbl 0309.55002 |

[7] | W. Thurston, The geometry and topology of \( 3\)-manifolds, Princeton Univ. preprint (1978). |

[8] | William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357 – 381. · Zbl 0496.57005 |

[9] | J. Weeks, Hyperbolic structures on three-manifolds, Princeton Ph.D. thesis (1985). |

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