Analysis and Geometry on Groups

Abstract

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.

Nash inequalities for finite Markov chains

“…Varopoulos 136~ (see also Ref. 37, Chs. 6 and 7) introduced these NashSobolev techniques in the study of random walk on finitely generated groups where they proved to be very effective.…”

A new characterization of convexity in free Carnot groups

“…. , F m } to n, there exists a (unique) Lie algebra morphism from f m,r to n extending L. The construction of such a Lie algebra f m,r is classical (see, e.g., [30,31]; the reader is also referred to [19,16] for the construction of a basis for f m,r ). We say that a Carnot group G is a free Carnot group if its Lie algebra is isomorphic to f m,r , for some m and r. Notice that, in this case, m necessarily equals the dimension of H (as in (2.1)) and r is the step of nilpotency of G.…”

A nonlinear PDE in mathematical finance

“…The Hörmander's result was the starting point of an extensive research aiming to investigate the regularity properties of the operators in (4) and their links with some suitable Lie group structures on R n . We refer to the paper by Rothschild and Stein [10] and to the book by Varopoulos, SaloffCoste and Coulhon [11] for a general regularity theory.…”