The Orbit Space of a Kleinian Group: Riley's Modest Example

In the previous paper, Robert Riley [4] and his computer file Poincaré found a fundamental domain for the action of a discrete group G of isometries of hyperbolic space H3 generated by three parabolics. In this paper, we show that the orbit space H3/G is homeomorphic to a complement 53 — k*, where k* is k union a point and where k is the (3,3,3) pretzel knot. Furthermore, HI 3/G is equipped with an infinite volume hyperbolic orbifold structure. This should not be confused with the complete finite volume hyperbolic structure on S3 — k. A neighborhood of k in H3/G is not the quotient of a horoball by a group of parabolic isometries with common fixed point. It is instead the quotient of a neighbourhood of the domain of discontinuity for G. In addition, G has elliptic elements of order three which give rise to three singular axes in the hyperbolic structure. These three axes meet at a point at infinity in H3/G. This accounts for the additional missing point in S3 — k*. A neighbourhood of this point is the quotient of a horoball by the Euclidean (3,3,3) triangle group. For a treatment of hyperbolic and orbifold structures, see Thurston [5]. The computer output form Riley's program consists of a picture of the fundamental domain 6D, and a data output giving the face pairings and face pairing transformations. For this paper, only the picture of <$ appearing in Section 3 of the preceding paper is used, as the information that it contains is sufficient to determine the face pairings uniquely. As a result, our labellings are different from those appearing in the data output, but the face pairings are the same. To show the homeomorphism H3/G = S3 — k*, we will glue up by identifying paired faces. The geometric structure will then arise as a direct consequence of the gluing. The domain <tf) is an infinite volume hyperbolic polyhedron. In the upper half space model %3 it lies between two EH-planes parallel to the imaginary axis of the boundary complex plane irQ. The E-closure of 6D, ÛD, contains three subsets of ir0. Two of these are compact, and are labelled Y and Z. The third region, labelled X, has connected closure in 770* and intersects any neighborhood of the point ( oo}. Figure 1 shows a slightly altered version of the original computer drawing with added labellings of faces and edges. Some of the edges of ty are EH-lines and, as such, they do not appear on the computer drawing. This poses no difficulty, as these edges will be subsumed by the first gluing step and will play no further part in the discussion.