Extending the Action of Schottky Groups on the Complex Anti-de Sitter Space to the Projective Space

Abstract

In this article we show that if a complex Schottky group, acting on the complex anti-de Sitter space, acts on the corresponding projective space as a Schottky group, then the space has signature (k, k): As a consequence, we are able to show the existence of complex Schottky groups, acting on $$ {\mathbb{P}}_\mathbb{C}^n $$, such that the complement of whose Kulkarni's limit set is not the largest open set on which the group acts properly and discontinuously. This is the starting point towards the understanding of the notion of the role of limit sets in the higher-dimensional setting.

This paper is dedicated to Pepe Seade in celebration of his 60th Birthday Anniversary.

Mathematics Subject Classification (2000). Primary 37F30, 32M05, 32M15; Secondary 30F40, 20H10, 57M60.