An introduction to asymptotic geometry

Abstract

This survey article presents the fundamentals of large-scale geometry of hyperbolic metric spaces and their boundaries. It is based on the book [S. Buyalo and V. Schroeder, Elements of asymptotic geometry. EMS Monographs in Mathematics. Zurich: European Mathematical Society (EMS). (2007; Zbl 1125.53036)]. A metric space X is Gromov hyperbolic if there exists δ≥0 such that for any four points x,y,z,w∈X the two largest of the three numbers |xy|+|zw|,|xz|+|yw|,|xw|+|yz| differ at most by 2δ, where |xy|, |zw| etc. denotes the distances. This can be interpreted by considering x,y,z,w as vertices of a complete graph K 4 in which the edges are assigned weights according to the distances in X. The δ-hyperbolicity condition says that one of three possible perfect matchings in this graph, the total lengths of two longest ones differ by at most 2δ. When X is a geodesic space, the above condition is equivalent to the δ-thinness of geodesic triangles. Since the survey is aimed at beginners in the subject, it illustrates the concepts with simple examples and provides motivation for the most important definitions. The topics include: geodesics and quasi-geodesics; Gromov product; Busemann functions; the boundary at infinity ∂ ∞ X and its metric structure. Furthermore, the author considers several classes of morphisms of hyperbolic spaces (bi-Lipschitz maps, rough isometries, rough similarities, and others). Additional topics include: quasi-metric spaces and their metric involutions; realization of a quasi-metric space as the boundary of a hyperbolic metric space.