Lectures on coarse geometry. (English) Zbl 1042.53027

Roe, John

University Lecture Series 31. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3332-4/pbk). vii, 175 p. (2003).

The author defines a coarse structure on a set \(X\) to be a collection of subsets of \(X\times X\) which contains the diagonal and is closed under taking subsets, inverses, products and finite unions. A set equipped with a course structures is called a coarse space. An example of a coarse structure associated to a metric space \(X\) is the collection of all subsets \(E \subset X\times X\) for which the projection maps \(\pi_1,\pi_2:E\to X\) are close. For the purpose of this review, the following informal definition is more useful: coarse geometry is the study of spaces (e.g. metric spaces) from a large scale point of view.
This book is divided into three parts. Part 1 (chapter 1 to 5) deals with the general theory of coarse structures, with a particular chapter on metric spaces. In particular, the author treats the subjects of growth, amenability and coarse cohomology. Part 2 of the book (chapters 6 to 8) deals with coarse negative curvature, limits of metric spaces and rigidity. In fact, Mostow’s rigidity theorem and Gromov’s large scale negative curvature are the two main motivations for the study of large scale geometry. Part 3 (chapters 9 to 11) contains recent results on asymptotic dimension and uniform embeddings into Hilbert spaces. The author establishes relations with property T and \(C^*-\)algebras. The general perspective on coarse structures that is presented in this book was first set out in the paper [K-Theory 11, 209–239 (1997; Zbl 0879.19003)] by N. Higson, E. K. Pedersen and the author. The subject matter and the treatment are very very interesting.

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

53C23	Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C24	Rigidity results
20F65	Geometric group theory
53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
46L85	Noncommutative topology
54E15	Uniform structures and generalizations
51K05	General theory of distance geometry

Keywords:

metric space; length space; hyperbolic space; property T; growth; metric tree; coarse structure; amenable space; amenable group; \(C^ *-\)algebra; coarse cohomology; negative curvature; limits of metric spaces; rigidity; asymptotic cone; ultralimit; Baum-Connes conjecture; Gromov hyperbolic; quasi-isometry; groupoid; asymptotic dimension

Citations:

Zbl 0879.19003