Discrete subgroups of complex hyperbolic motions (Q1385165)

The two-dimensional complex hyperbolic space \(CH^2\) is taken as the subset of the projective space \(CP^2\) defined by \(\langle z,z\rangle<0\), where \(\langle , \rangle\) is an indefinite Hermitean product. The paper derives necessary conditions on two elements of the holomorphic isometry group \(PU(2,1)\) to generate a discrete subgroup. Moreover, the authors relate the Carnot-Carathéodory distance and the Cygan distance on the boundary of \(CH^2\) by some explicit formula. They derive a compactness theorem, called the stable basin theorem, for elements close to the identity in \(PU(2,1)\), and they supply conditions for determining when two elements of \(PU(2,1)\) commute in terms of their action on the boundary.