Hyperbolic Structures of Arithmetic Type on Some Link Complements

Abstract

We consider here the Bianchi groups PSL2 (0d), where (9d is the ring of integers in the imaginary quadratic field Q{J — d). In the five cases d = 1,2,3,7,11 when &d is Euclidean, there is a PSL2(0d)-invariant tesselation 9~d of hyperbolic 3-space H 3 by congruent ideal polyhedra (that is, polyhedra with all vertices at infinity). The definition of ^ is very simple: its vertices are the points of-d) u {00} c = CP 1 = dH 3, and each a ePSL2(0d) determines an edge which is the geodesic in H 3 having endpoints a//? and y/d. The tesselations 5 ^ and ^ are by regular ideal octahedra and tetrahedra, respectively, and are classical objects; for example, & \\ is constructed in [2]. The tesselations «^, by ideal cuboctahedra, and ST-i, by ideal triangular prisms, are implicit in Chapter 6 of [9]; ^ is a tesselation by ideal truncated tetrahedra. The tesselations ZTd have been described also in [4] and in the recent Oxford thesis of J. Cremona. In each case the full group of orientation-preserving combinatorial symmetries of ZTd is PGL2 {(9d). It follows that hyperbolic 3-manifolds of the form H 3 /T for T a torsion-free subgroup of PGL2 (0d) can be characterized purely combinatorially by the condition that they can be decomposed into ideal polyhedra in a fashion locally