### Impacto

Montesinos Amilibia, José María and Hilden, Hugh Michael and Lozano Imízcoz, María Teresa and Whitten, Wilbur Carrington
(1987)
*On universal groups and three-manifolds.*
Inventiones Mathematicae, 87
(3).
pp. 441-456.
ISSN 0020-9910

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## Abstract

Let P be a regular dodecahedron in the hyperbolic 3-space H3with the dihedral angles 90∘. Choose 6 mutually disjoint edgesE1,E2,⋯,E6 of P such that each face of P intersects E1∪E2∪⋯∪E6 in one edge and the opposite vertex. Let U be the group of orientation-preserving isometries of H3 generated by 90∘-rotations about E1,⋯,E6. It was observed by W. Thurston that H3/U=S3 and that the projection H3→H3/U is a covering branched over the Borromean rings with branching indices 4. The main result of the paper is the following universality of U. Theorem: For every closed, oriented 3-manifold M there exists a subgroup G of U of finite index such that M=H3/G. In other words M is a hyperbolic orbifold finitely covering the hyperbolic orbifold H3/U.

The main ingredient of the proof is the strict form of the universality of the Borromean rings (earlier obtained by the first three authors): each closed, oriented 3-manifold is shown to be a covering of S3 branched over the Borromean rings with indices 1, 2, 4.

This theorem offers a new approach to the Poincaré conjecture: If M=H3/G as above and π1(M)=1 then G is generated by elements of finite order. The authors start off an algebraic investigation of U⊂PSL2(C) by constructing three generators of U which are 2×2 matrices over the ring of algebraic integers in the field Q(2√,3√,5√,1√+5√,−1−−−√).

Item Type: | Article |
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Uncontrolled Keywords: | regular dodecahedron; hyperbolic 3-space; covering branched over the Borromean rings; 3-manifold; hyperbolic orbifold; Poincaré conjecture; PSL 2 (bbfC) |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 17162 |

References: | Armstrong, M.A.: The fundamental group of the orbit space of a discontinuous group. Proc. Camb. Phil. Soc.64, 299–301 (1968) Fox, R.H.: A note on branched cyclic coverings of spheres. Rev. Mat. Hisp.-Am.32, 158–166 (1972) Fox, R.H.: Covering spaces with singularities. Algebraic geometry and topology. A symposium in honor of S. Lefschetz. Princeton 1957 Hilden, H.: Every closed, orientable 3-manifold is a 3-fold branched covering space ofS 3. Bull. Am. Math. Soc.80, 1243–1244 (1974) Hilden, H., Lozano, M., Montesinos, J.: The Whitehead link, the Borromean rings and the Knot 946 are universal. Collect. Math.34, 19–28 (1983) Hilden, H., Lozano, M., Montesinos, J.: Universal knots. Lect. Notes Math.1144 (D. Rolfsen (ed.)) 1985 Hilden, H., Lozano, M., Montesinos, J.: On knots that are universal. Topology24, 499–504 (1985) Hirsch, U.: Über offene Abbildungen auf die 3-sphäre. Math. Z.140, 203–230 (1974) Morgan, J.W., Bass, H.: The Smith Conjecture. Academic Press, 1984 Montesinos, J.: A representation of closed, orientable 3-manifolds as 3-fold branched coverings ofS 3. Bull. Am. Math. Soc.80, 845–846 (1974) Montesinos, J.: Sobre la Conjetura de Poincaré y los recubridores ramificades sobre un nudo. Ph. D. Thesis, Madrid, 1971 Montesinos, J.: Una nota a un teorema de Alexander. Rev. Mat. Hisp.-Am.32, 167–187 (1972) Thurston, W.: Universal Links (preprint, 1982) Thurston, W.: The geometry and topology of three-manifolds. Princeton University Press (to appear 1977/1978) |

Deposited On: | 22 Nov 2012 10:20 |

Last Modified: | 07 Feb 2014 09:43 |

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