Lectures on Coarse Geometry
The book is based on lectures given by the author at the Penn State university. The notion of ‘coarse geometry’ was invented to keep track of large scale properties of metric spaces. The first part of the book introduces an abstract notion of a coarse structure, a notion of a bounded geometry coarse space, its growth and a notion of an amenable metric space, and discusses coarse algebraic topology. The main topic in the middle part is the Mostow rigidity theorem saying that if two compact hyperbolic manifolds of dimensions at least 3 are homotopy equivalent, they are isometric. The last part of the book contains a discussion of a notion of asymptotic dimensions and uniform embeddings into Hilbert spaces, together with relations to the Kazhdan property of discrete groups. The book offers a very readable description of a circle of ideas around the notion of coarse geometry.