Exceptional Regions and Associated Exceptional Hyperbolic 3-Manifolds

We investigate the seven exceptional families as defined in [GMT]. Experimental as well as rigorous evidence suggests that to each family corresponds exactly one manifold. A certain two generator subgroup in PSL(2,C) is specified for each of the seven families in [GMT]. Using Newton’s method for finding roots of polynomials in several variables we solve the relation equations specifying the generators to high precision. Then, using the LLL algorithm [Neum] we find exact entries of the generating matrices and in all cases verify with exact arithmetic that they satisfy the relations. This procedure allows us to compute the invariant trace fields [Neum] associated with the conjectured manifolds. In part, our results provide a verification of earlier results of K. Jones and A. Reid [JR] which were obtained by arithmetic methods. We carry out a search of the census of hyperbolic manifolds given in SnapPea and find hyperbolic manifolds with fundamental groups isomorphic to some of subgroups mentioned above. In addition, we obtain results on X3 and X4 which are not discussed in the K. Jones and A. Reid paper.