Analysis and Geometry on Groups

Includes Bibliography (148-155) and Index

1. Introduction
2. Dimensional inequalities for semigroups of operators on the Lp spaces
3. Systems of vector fields satisfying Hr̲mander's condition
4. The heat kernel on nilpotent Lie groups
5. Local theory for sums of squares of vector fields
6. Convolution powers on finitely generated groups
7. Convolution powers on unimodular compactly generated groups
8. The heat kernel on unimodular Lie groups
9. Sobolev inequalities on non-unimodular Lie groups
10. Geometric applications

The geometry and analysis that is discussed in this book extends to classical results for general discrete or Lie groups, and the methods used are analytical but have little to do with what is described these days as real analysis. Most of the results described in this book have a dual formulation; they have a 'discrete version' related to a finitely generated discrete group, and a continuous version related to a Lie group. The authors chose to centre this book around Lie groups but could quite easily have pushed it in several other directions as it interacts with opetators, and probability theory, as well as with group theory. This book will serve as an excellent basis for graduate courses in Lie groups, Markov chains or potential theory.