Spherical CR Geometry and Dehn Surgery,

There are several ways to generalise the hyperbolic plane and its isometry group to objects in higher dimensions. Perhaps the most familiar is (real) hyperbolic three space, popularised by the work of Thurston [14]. The Poincaré disc and half plane models of the hyperbolic plane naturally come with a complex structure and it is natural to generalise them to complex hyperbolic space in higher complex dimensions; see [4] or [8] for further details. A useful model for complex hyperbolic space is the unit ball in Cn equipped with the Bergman metric. When n = 1 this is just the Poincaré metric on the unit disc in C. When n ≥ 2 complex hyperbolic space does not have constant curvature, but has pinched negative curvature, which we normalise to lie between −1 and −1/4. From now on we concentrate on the case n = 2. The hyperbolic plane is isometrically embedded into complex hyperbolic two-space HC in two geometrically distinct ways. First, the intersection of the unit ball in C2 with a complex line (for example one of the complex coordinate axes) is a totally geodesic disc. The restriction of the Bergman metric to this disc is the Poincaré metric with constant curvature −1. On the other hand, the intersection of HC with a Lagrangian plane (for example the collection of points with real coordinates) is also a totally geodesic disc. In this case, the restriction of the Bergman metric is the Klein metric on the hyperbolic plane with constant curvature −1/4. The group of holomorphic isometries of HC is the projective unitary group PU(2, 1). It s often useful to lift to the matrix group SU(2, 1), which is a threefold cover of PU(2, 1). Non-trivial elements of PU(2, 1) fall into the three classes familiar from real hyperbolic geometry. Namely, A ∈ PU(2, 1) is loxodromic if it fixes exactly two points of ∂HC, one of which is attractive and the other repulsive; A is parabolic if it fixes exactly one point of ∂HC and is elliptic if it fixes at least one point of HC. Elliptic isometries are either a complex reflection fixing a point or a complex line, or else are called regular. Complex reflections correspond to matrices in SU(2, 1) with a repeated eigenvalue and regular elliptic maps correspond to matrices with distinct eigenvalues. The full group of complex hyperbolic isometries P̂U(2, 1) is generated by PU(2, 1) and an antiholomorphic reflection fixing a Lagrangian plane. An example of such an involution is complex conjugation