Publication: On the universal group of the Borromean rings
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1988
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Springer-Verlag
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The authors improve the result of their previous paper on universal groups [the authors and W. Whitten, Invent. Math. 87, 411-456] and apply them to prove several interesting results on 3-manifolds. We quote some of these results below, adding necessary definitions: Definition. Let U be a discrete group of isometries of hyperbolic 3-space, H 3 . One says that U is universal if it has the following property: If M 3 is any closed oriented 3- manifold, then there is a finite index subgroup, G(M 3 ), of U such that M 3 is the orbit space of the action of G(M 3 ) on H 3 . Theorem 1. There is a universal group U which is a subgroup of PSL 2 (A ^), where A ^ is the ring of algebraic integers of the field Q(2,i,t). Furthermore U is an arithmetic group (a subgroup of index 120 in the tetrahedral reflection group). Theorem 4. The universal group U has an index four subgroup N which acts freely on H 3 . Also, U/N is cyclic. Theorem 5. Every closed oriented 3- manifold can be “pentagulated”; that is, obtained from a finite set of dodecahedra by pasting along pentagonal faces in pairs. Theorem 6. Any closed oriented 3-manifold has a cell decomposition whose 2-skeleton is the image of an immersion of a disconnected surface with boundary. The immersion is in general position. Definition. A 3-manifold is called dodecahedral if it is a complete hyperbolic 3-manifold with a tesselation by regular, right-dihedral angled hyperbolic dodecahedra. Theorem 7. Every closed 3-manifold is the orbit space of an orientation preserving ℤ/4 action on a dodecahedral manifold. Theorem 8. Let π be the fundamental group of a compact oriented 3-manifold M 3 . Then π is isomorphic to a group of fixed point free, tesselation preserving, isometries of a dodecahedral manifold.
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Proceedings of the Second Topology Symposium, held in Siegen, FRG, Jul. 27–Aug. 1, 1987
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H.M. Hilden, M.T. Lozano, J.M. Montesinos, and W. Whitten, "On universal groups and three-manifolds", Inventiones, vol. 87 (1987), 441–456.
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Hyman Bass, "Groups of Integral Representation Type", Pacific J. Math., vol. 86 (1980), 15–51.
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H. Hilden, M. Lozano and J. Montesinos, "The Whitehead link, the Borromean rings and the Knot 946 are universal", Collec. Math., 34 (1983), 19–28.
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H. Hilden, M. Lozano and J. Montesinos, "On knots that are universal", Topology, vol. 24 (1985), 499–504.
Alan W. Reid, "Arithmetic Kleinian groups and their Fuchsian subgroups", Ph. D. thesis, Univ. of Aberdeen, Scotland, 1987.
E.B. Vinberg, "Discrete Groups generated by reflections in Lobacerskii spaces", Mat. Sbornik 114 (1967), 471–488. (A.M.S. translation, Math. USSR Sbornik 1 (1967), 429–444.)