An Invitation to Hyperbolic Groups

The topic of this bookare hyperbolic groups in the sense of Gromov. These groups form one of the fundamental examples of study in geometric group theory. Fundamental examples include free groups and the fundamental groups of closed negatively curved manifolds. Hyperbolic groups capture the notion of negative curvature for discrete groups. This book starts with the basics, defining the groups and metric spaces of interest, and focussing on key examples (trees and free groups, and the hyperbolic plane and surface groups). It goes through the basic theory, providing an introduction to the basic ideas, results and constructions. In the second part of the book, more advanced and recent topicsare covered, in an attempt to bring the reader near to the modern research environment. A discussion of more general notions (coming from actions on delta-hyperbolic groups that are not proper and cocompact)is provided, and an indication of important and profitable directions for the futureis given.