International Congress of Mathematicians

340 Richard Evan SchwartzThe reflection triangle groups are rigid in Isom(iì 3 ), in the sense that any twodiscrete embeddings of the same group are conjugate. We are going to replace H 3by CH", the complex hyperbolic plane. In this case, we get nontrivial deformations.These deformations provide an attractive problem, because they furnish some of thesimplest interesting examples in the still mysterious subject of complex hyperbolicdeformations. While some progress has been made in understanding these examples,there is still a lot unknown about them.In §2 we will give a rapid introduction to complex hyperbolic geometry. In§3 we will explain how to generate some complex hyperbolic triangle groups. In §4we will survey some results about these groups and in §5 we will present a morecomplete conjectural picture. In §6 we will indicate some of the techniques we usedin proving our results.2. The complex hyperbolic planeThe book [8] is an excellent general reference for complex hyperbolic geometry.Here are some of the basics.C 2 ' 1 is a copy of the vector space C 3 equipped with the Hermitian formn(U, V) = -u 3 v 3 + Y^ ujVj- (1)Here U = (u\,u 2 ,uz) and V = (v\,v 2 ,vz). A vector V is called negative, null, orpositive depending (in the obvious way) on the sign of (V, V). We denote the set ofnegative, null, and positive vectors, by A r _, N 0 and N + respectively.C" includes in complex projective space CP" as the affine patch of vectorswith nonzero last coordinate. Let [ ] : C 2 ' 1 — {0} —ï CP 2 be the projectivizationwhose formula, expressed in the affine patch, is3=1[(vi,v 2 ,v 3 )] = (vi/v 3 ,v 2 /v 3 ). (2)The complex hyperbolic plane, CH", is the projective image of the set of negativevectors in C 2 ' 1 . That is, CH 2 = [AT_]. The ideal boundary of CH 2 is the unitsphere S 3 = [N 0 ]- If [X],[F] £ CH n the complex hyperbolic distance ß([X],[F])satisfiesQ([X],[Y}) = 2cosh- 1 y/6(X7ni 6(X,Y) = j ^ ^ y j - (3)Here X and Y are arbitrary lifts of [X] and [Y]. See [8, 77]. The distance we definedis induced by an invariant Riemannian metric of sectional curvature pinched between— 1 and —4. This Riemannian metric is the real part of a Kahler metric.SU(2,1) is the Lie group of ( , ) preserving complex linear transformations.PU(2,1) is the projectivization of SU(2,1) and acts isometrically on CH". Themap SU(2,1) —t PU(2,1) is a 3-to-l Lie group homomorphism. The group ofholomorphic isometries of CH" is exactly PU(2,1). The full group of isometries