An Overview of Complex Kleinian Groups

Abstract

Dedicated to Vale, on the occasion of his 65th birthday.Abstract. Classical Kleinian groups are discrete subgroups of P SL(2, C) acting on the complex projective line P 1 C , which actually coincides with the Riemann sphere, with non-empty region of discontinuity. These can also be regarded as the monodromy groups of certain differential equations. These groups have played a major role in many aspects of mathematics for decades, and also in physics. It is thus natural to study discrete subgroups of the projective group P SL(n, C), n > 2. Surprisingly, this is a branch of mathematics which is in its childhood, and in this article we give an overview of it.1991 Mathematics Subject Classification. Primary: 3nQ45, 37F45; Secondary nnE40, 57R30.