International Congress of Mathematicians

Complex Hyperbolic Triangle Groups 341of CH" is generated by PU(2,1) and by the antiholomorphic map (01,02,23) —^(~Zi,Z 2 ,Z 3 ).An element of PU(2,1) is called elliptic if it has a fixed point in CH". Itis called hyperbolic (or loxodromic) if there is some e > 0 such that every point inCH" is moved at least e by the isometry. An element which is neither elliptic norhyperbolic is called parabolic.CH" has two different kinds of totally geodesic subspaces, real slices andcomplex slices. Every real slice is isometric to CH" n R" and every complex slice isisometric to CH 2 n C 1 . The ideal boundaries of real and complex slices are called,respectively, H-circles and C-circles. The complex slices naturally implement thePoincaré model of the hyperbolic plane and the real slices naturally model the Kleinmodel. It is a beautiful feature of the complex hyperbolic plane that it containsboth models of the hyperbolic plane.3. Reflection triangle groupsThere are two kinds of reflections in lsom(CH"). A real reflection is ananti-holomorphic isometry conjugate to the map (z,w) —¥ (z,w). The fixed pointset of a real reflection is a real slice. We shall not have much to say about theexplicit computation of real reflections, but rather will concentrate on the complexreflections.A complex reflection is a holomorphic isometry conjugate to the involution(z,w) —¥ (z,—w). The fixed point set of a complex reflection is a complex slice.There is a simple formula for the general complex reflection: Let C £ N + . Givenany U £ C 2 ' 1 define2(U,C) dIc(U) = ^U + -çëjçfC. (4)le is a complex reflection.We also have the formulaU M V = («3W2 — «2^3, U1V3 — U3V1, U1V2 — u 2 vi). (5)This vector is such that (U, U S V) = (V, U S V) = 0. See [8, p. 45].Equations 4 and 5 can be used in tandem to rapidly generate triangle groupsdefined by complex reflections. One picks three vectors Vi,V2,V2 £ AT_. Next, welet Cj = Vj-i M Vj+i- Indices are taken mod 3. Finally, we let fi = Ic r Thecomplex reflection fi fixes the complex line determined by the points [Yfi-i] and[Vj + i]. This, the group (Ii,l2,h) is a complex-reflection triangle group determinedby the triangle with vertices [Vi], [V2], [V3].Here is a quick dimension count for the space of (p, q, r)-triangle groups generatedby complex reflections. We can normalize so that [Vi] = 0. The stabilizerof 0 in PU(2,1) acts transitively on the unit tangent space at 0. We cantherefore normalize so that [V2] = (s,0) where s £ (0,1). Finally, the isometries(z, w) —¥ (z, exp(i9)iv) stabilize both [Vi] and [V2]. Applying a suitable isometry wearrange that [V3] = (t+iu, v) where t,u,v £ (0,1). We cannot make any further normalizations,so the space of triangles in CH" mod isometry is 4-real dimensional.