On universal groups and three-manifolds

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−−−√).