##
**Analysis on Lie groups.**
*(English)*
Zbl 0813.22003

Cambridge Tracts in Mathematics. 100. Cambridge etc.: Cambridge University Press. xii, 156 p. (1992).

This is a useful exposition of extensive work by the authors, scattered in many papers over the past decade, involving subjects of classical analysis such as Sobolev, Harnack and Dirichlet inequalities, heat diffusion semigroup, random walk, convolution operators, Dirichlet spaces in the sense of Beurling & Deny, …in the context of (possibly discrete) groups.

A natural distance on a finitely generated discrete group \(G\) arises as follows: Let \(g_ 1,\dots,g_ k\) be generators of \(G\); then the ball of radius \(n\) centered at the identity of \(G\) is the set of group elements \[ \{g^{\varepsilon_ 1}_{i_ 1} \cdots g^{\varepsilon_ p}_{i_ p} : 0 \leq p \leq n,\;i_ \alpha = 1,\dots,k,\;\varepsilon_ \alpha = \pm1\}. \] Left translation on \(G\) gives the distance between any two elements of \(G\). The number \(\gamma(n)\) of group elements lying in the above \(n\)-ball is the growth function of \(G\), studied by M. Gromov and others. The distinction between exponential and polynomial growth is nearly a dichotomy.

Such notions of distance and volume growth can be extended in a simple way to the case of a locally compact group \(G\) generated by a symmetric, compact neighborhood of the identity. This yields Sobolev and Dirichlet norms in finitely generated groups and one of the main theorems (Chapter VI) asserts for finitely generated, discrete, superpolynomial (in an obvious sense) groups \(\Gamma\) the equivalence of Dirichlet type inequality and decay of the form \(\nu^{(k)}(e) = {\mathcal O}(k^{-A})\) for the convolution powers \(\nu^{(k)}\) of a probability measure \(\nu\) on \(\Gamma\).

Chapter VII is devoted in part to the difficult task of extending this result to locally compact, unimodular, compactly generated groups. Chapters II and III deal with numerous results and techniques concerning semigroups, Laplacians and the Carnot-Carathéodory distance associated with vector fields satisfying Hörmander’s condition. Chapter IV uses the results of Chapters II and III in order to learn “almost everything about the heat kernel and the Sobolev inequalities associated to a family of Hörmander fields on a nilpotent Lie group.” In Chapter VIII the heat kernel on a unimodular Lie group is estimated with respect to volume growth, polynomial and exponential, the aim being to get a gaussian factor in the upper and lower estimates. Chapter IX is entitled “Sobolev inequalities on non-unimodular Lie groups”. The final chapter is a succinct account of several geometric applications such as the transience and recurrence of Brownian motion on covering manifolds. There are many open questions in the references and comments at the end of each chapter.

Clearly, a tremendous amount of effort has gone into organising the present theory, involving interlocking disciplines as it does, into a readable unity.

A natural distance on a finitely generated discrete group \(G\) arises as follows: Let \(g_ 1,\dots,g_ k\) be generators of \(G\); then the ball of radius \(n\) centered at the identity of \(G\) is the set of group elements \[ \{g^{\varepsilon_ 1}_{i_ 1} \cdots g^{\varepsilon_ p}_{i_ p} : 0 \leq p \leq n,\;i_ \alpha = 1,\dots,k,\;\varepsilon_ \alpha = \pm1\}. \] Left translation on \(G\) gives the distance between any two elements of \(G\). The number \(\gamma(n)\) of group elements lying in the above \(n\)-ball is the growth function of \(G\), studied by M. Gromov and others. The distinction between exponential and polynomial growth is nearly a dichotomy.

Such notions of distance and volume growth can be extended in a simple way to the case of a locally compact group \(G\) generated by a symmetric, compact neighborhood of the identity. This yields Sobolev and Dirichlet norms in finitely generated groups and one of the main theorems (Chapter VI) asserts for finitely generated, discrete, superpolynomial (in an obvious sense) groups \(\Gamma\) the equivalence of Dirichlet type inequality and decay of the form \(\nu^{(k)}(e) = {\mathcal O}(k^{-A})\) for the convolution powers \(\nu^{(k)}\) of a probability measure \(\nu\) on \(\Gamma\).

Chapter VII is devoted in part to the difficult task of extending this result to locally compact, unimodular, compactly generated groups. Chapters II and III deal with numerous results and techniques concerning semigroups, Laplacians and the Carnot-Carathéodory distance associated with vector fields satisfying Hörmander’s condition. Chapter IV uses the results of Chapters II and III in order to learn “almost everything about the heat kernel and the Sobolev inequalities associated to a family of Hörmander fields on a nilpotent Lie group.” In Chapter VIII the heat kernel on a unimodular Lie group is estimated with respect to volume growth, polynomial and exponential, the aim being to get a gaussian factor in the upper and lower estimates. Chapter IX is entitled “Sobolev inequalities on non-unimodular Lie groups”. The final chapter is a succinct account of several geometric applications such as the transience and recurrence of Brownian motion on covering manifolds. There are many open questions in the references and comments at the end of each chapter.

Clearly, a tremendous amount of effort has gone into organising the present theory, involving interlocking disciplines as it does, into a readable unity.

Reviewer: E.J.Akutowicz (Montpellier)

### MSC:

22E30 | Analysis on real and complex Lie groups |

43A80 | Analysis on other specific Lie groups |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

22E40 | Discrete subgroups of Lie groups |

35K05 | Heat equation |

20F05 | Generators, relations, and presentations of groups |

20F65 | Geometric group theory |