Geometry of Kleinian groups and its applications

Abstract

The notion of Kleinian groups was first introduced by Poincaré in the 19th century, and their study from a viewpoint of complex analysis was developed by Ahlfors, Bers, Kra, Maskit and Marden among others. After Thurston’s innovative work, topological study of Kleinian groups was started and flourished. In this survey article, we look back on Thurston’s influential work and his famous 10 open questions, and explain their significance and how they have been solved. In the latter half, we shall look at more recent development of the theory of Kleinian groups.