Lectures on Coarse Geometry
About this Title
John Roe, Pennsylvania State University, University Park, PA
Publication: University Lecture Series
Publication Year: 2003; Volume 31
ISBNs: 978-0-8218-3332-2 (print); 978-1-4704-2176-2 (online)
MathSciNet review: MR2007488
MSC: Primary 53C24; Secondary 20F65, 46L05, 51K05, 54E25
Coarse geometry is the study of spaces (particularly metric spaces) from a ‘large scale’ point of view, so that two spaces that look the same from a great distance are actually equivalent. This point of view is effective because it is often true that the relevant geometric properties of metric spaces are determined by their coarse geometry. Two examples of important uses of coarse geometry are Gromov's beautiful notion of a hyperbolic group and Mostow's proof of his famous rigidity theorem.
The first few chapters of the book provide a general perspective on coarse structures. Even when only metric coarse structures are in view, the abstract framework brings the same simplification as does the passage from epsilons and deltas to open sets when speaking of continuity.
The middle section reviews notions of negative curvature and rigidity. Modern interest in large scale geometry derives in large part from Mostow's rigidity theorem and from Gromov's subsequent ‘large scale’ rendition of the crucial properties of negatively curved spaces.
The final chapters discuss recent results on asymptotic dimension and uniform embeddings into Hilbert space.
John Roe is known for his work on index theory, coarse geometry and topology. His exposition is clear and direct, bringing insight to this modern field of mathematics. Students and researchers who wish to learn about contemporary methods of understanding the geometry and topology of manifolds will be well served by reading this book.
Graduate students and research mathematicians interested in geometry, topology, and index theory.
Table of Contents
- Chapter 1. Metric spaces
- Chapter 2. Coarse spaces
- Chapter 3. Growth and amenability
- Chapter 4. Translation algebras
- Chapter 5. Coarse algebraic topology
- Chapter 6. Coarse negative curvature
- Chapter 7. Limits of metric spaces
- Chapter 8. Rigidity
- Chapter 9. Asymptotic dimension
- Chapter 10. Groupoids and coarse geometry
- Chapter 11. Coarse embeddability