Elements of Asymptotic Geometry
This book is devoted to a study of the generalisation of classical hyperbolic geometry developed by M. Gromov. It is based on lectures given at St. Petersburg and Zürich. The classical hyperbolic space Hn of dimension n can be realised as a ball in Rn and there is an induced conformal structure on the sphere forming its boundary. Isometries of Hn then induce conformal maps on the boundary. Gromov noticed that there is a specific inequality in terms of mutual distances of four points in Hn, which captures essential asymptotic properties of the hyperbolic space. The inequality is then used as a key tool in a definition of generalised hyperbolic spaces studied in the book. In the first part, the authors describe properties of the Gromov hyperbolic spaces in a close analogy with the homogeneous model Hn. In the second part, more general tools are used and some applications are given (e.g. non-existence theorem for embeddings). Bibliographic notes at the end of each chapter are quite useful.