# On the universal group of the Borromean rings

### Impacto

Montesinos Amilibia, José María and Hilden, Hugh Michael and Lozano Imízcoz, María Teresa (1988) On the universal group of the Borromean rings. In Differential topology. Lecture notes in mathematics (1350). Springer-Verlag, Berlín, pp. 1-13. ISBN 0075-8434

## Abstract

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.

Item Type: Book Section Proceedings of the Second Topology Symposium, held in Siegen, FRG, Jul. 27–Aug. 1, 1987 discrete group of isometries of hyperbolic 3-space; closed oriented 3- manifold; universal group; ring of algebraic integers; arithmetic group; dodecahedra; immersion; complete hyperbolic 3-manifold; orbit space; dodecahedral manifold; fundamental group Sciences > Mathematics > Topology 17096 15 Nov 2012 09:31 12 Dec 2018 15:13