The arithmetic structure of a universal group

Abstract

Let H3 denote hyperbolic 3-space and identify its group of orientation-preserving isometries with PSL(2,C). In the context of usage here, a universal group is a cocompact subgroup U of PSL(2,C) which has the property that every closed, oriented 3-manifold M is homeomorphic to the quotient H3/G for some finite-index subgroup G of U. In other words, every closed, oriented 3-manifold M is the underlying space of some finite orbifold cover of H3/U. The authors of this paper, jointly with W. C. Whitten, previously constructed a particular group U0 which they showed to be universal [Invent. Math. 87 (1987), no. 3, 441–456;]. In this work, they demonstrate that U0 is in fact arithmetic. Specifically, let F denote the unique quartic field of discriminant −400 (which has one complex place) and let A denote the quaternion algebra over F which is ramified only at the two real places. They show that A contains a particular order whose group of elements of norm one is commensurable with U0. It is suggested that the arithmetic structure may be of use in studying this group (in particular, the finite index subgroups of U0 generated by elliptic elements are relevant to the Poincaré conjecture). They further conjecture that this is the simplest such example: more precisely, they conjecture that 400 is the smallest absolute value of the invariant trace field discriminant of any universal arithmetic group.