Hyperbolic Structures of Arithmetic Type on Some Link Complements

-d) u {00} c= CP = dH, and each a ePSL2(0d) determines an edge which is the geodesic in H 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 /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 isomorphic to 2Td. As examples, following a technique of [9] we construct hyperbolic structures on a number of link complements in S. Many of these were known before; see [5, 6,9,10]. By a recent result of [3], there are only finitely many numbers d for which subgroups of PSL2(0d) can yield hyperbolic structures on link complements. But it seems that the only ones presently known to yield link complements are d = 1, 2, 3, 7,11. There is a more general definition of a PSL2 (0d)-invariant tesselation ^ of H 3