Compact quotient manifolds of domains in a complex 3-dimensional projective space and the Lebesgue measure of limit sets (Q1922159)

The author considers groups acting on the three-dimensional complex projective space \(P^3\) which are the analogs of function groups in the classical theory of Kleinian groups. More precisely, he considers discontinuous groups \(\Gamma< PGL_4 (\mathbb{C})\) acting discontinuously and freely on some domain \(\Omega \subset P^3\) so that \(\Omega/ \Gamma\) is a compact manifold. In the above situation, the domain \(\Omega\) is assumed to contain a subdomain biholomorphically equivalent to an open set of \(P^3\) containing a complex projective line \(P^1\). Two main tools in the theory of Kleinian groups are the Maskit combination theorems, which permit to obtain new Kleinian groups from old ones. One of the main properties of these combination theorems is the fact that the combination of Kleinian groups with Lebesgue measure zero limit sets gives again Kleinian groups with Lebesgue measure zero limit sets. The main result of this paper is the validity of the above property for the groups acting on \(P^3\) as above.