On the universal group of the Borromean rings

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.