Ergodic theory and semisimple groups.

*(English)*Zbl 0571.58015
Monographs in Mathematics, Vol. 81. Boston-Basel-Stuttgart: BirkhĂ¤user. X, 209 p. DM 89.00 (1984).

A class of discrete subgroups of Lie groups, called lattices (namely, those for which the corresponding homogeneous space admits a finite invariant measure), play an important role in various branches of mathematics such as number theory, geometry, dynamics, ergodic theory etc. In recent years there has been remarkable progress in the understanding of these subgroups, thanks to the profound work of G. A. Margulis on ”irreducible” lattices in semisimple Lie groups with trivial center and \({\mathbb R}\)-rank at least 2. The theorems on superrigidity (about extending a finite dimensional representation of the lattice to the whole group), arithmeticity (showing that the lattice can be viewed as arising from a well-known arithmetical construction) and on normal subgroups of such lattices are some of the high points.

To put this in a proper perspective for a reader not familiar with the area, it may be noted that various aspects of lattices in solvable Lie groups have been satisfactorily understood for quite some time, through the work of L. Auslander, G. D. Mostow and various other authors (for details see M. S. Raghunathan’s book [Discrete subgroups of Lie groups. Berlin etc.: Springer (1972; Zbl 0254.22005)].

Ergodic theory plays a role in some of Margulis’ proofs and especially in a simplification made by H. Furstenberg. On the other hand, the present author discovered that the rigidity theory could be recast in a wider framework to study ergodic actions with a finite invariant measure, of the semisimple Lie groups as above, proving in particular, an interesting and rather unexpected rigidity property of such actions (asserting that ”orbit equivalence” of such actions, of possibly different groups, already implies conjugacy of the actions).

The book under review gives a lucid account of the work of Margulis and the subsequent developments. There is a major hurdle involved in such a task. The work involves ideas both from the theory of algebraic groups and ergodic theory, areas which are traditionally pursued by almost disjoint sets of mathematicians. The author has made an attempt to be accessible to both the groups. While the relevant ergodic theory, including C. C. Moore’s ergodicity theorem, is recalled with adequate proofs, the arguments from algebraic group theory are illustrated by simple examples, typically \(\mathrm{SL}(n,\mathbb R)\), for those not comfortable with the general theory. The approach has contributed a good deal in making the book readable.

Apart from the topics mentioned above the work also involves study of amenable actions (this is a generalization of the classical notion of amenable groups) and the Kazhdan property. The author also gives a no- tears exposition of these topics. The topics are of independent interest; the Kazhdan property, for instance, is involved in something as diverse as the study of isometry-invariant finitely additive measures on spheres [see S. Wagon, The Banach-Tarski paradox. Cambridge etc.: Cambridge University Press (1985; Zbl 0569.43001)]. The chapters on these topics might also benefit many mathematicians not directly concerned with the main theme of the book.

This reviewer would have been happier if the chapter on ergodicity were to cover a few more things that would connect it with the classical ergodic theory, though they are not needed in the subsequent chapters. However, this is a minor point. The author should be thanked for the book, the first of its kind, which would go a long way in generating interest in the subject.

To put this in a proper perspective for a reader not familiar with the area, it may be noted that various aspects of lattices in solvable Lie groups have been satisfactorily understood for quite some time, through the work of L. Auslander, G. D. Mostow and various other authors (for details see M. S. Raghunathan’s book [Discrete subgroups of Lie groups. Berlin etc.: Springer (1972; Zbl 0254.22005)].

Ergodic theory plays a role in some of Margulis’ proofs and especially in a simplification made by H. Furstenberg. On the other hand, the present author discovered that the rigidity theory could be recast in a wider framework to study ergodic actions with a finite invariant measure, of the semisimple Lie groups as above, proving in particular, an interesting and rather unexpected rigidity property of such actions (asserting that ”orbit equivalence” of such actions, of possibly different groups, already implies conjugacy of the actions).

The book under review gives a lucid account of the work of Margulis and the subsequent developments. There is a major hurdle involved in such a task. The work involves ideas both from the theory of algebraic groups and ergodic theory, areas which are traditionally pursued by almost disjoint sets of mathematicians. The author has made an attempt to be accessible to both the groups. While the relevant ergodic theory, including C. C. Moore’s ergodicity theorem, is recalled with adequate proofs, the arguments from algebraic group theory are illustrated by simple examples, typically \(\mathrm{SL}(n,\mathbb R)\), for those not comfortable with the general theory. The approach has contributed a good deal in making the book readable.

Apart from the topics mentioned above the work also involves study of amenable actions (this is a generalization of the classical notion of amenable groups) and the Kazhdan property. The author also gives a no- tears exposition of these topics. The topics are of independent interest; the Kazhdan property, for instance, is involved in something as diverse as the study of isometry-invariant finitely additive measures on spheres [see S. Wagon, The Banach-Tarski paradox. Cambridge etc.: Cambridge University Press (1985; Zbl 0569.43001)]. The chapters on these topics might also benefit many mathematicians not directly concerned with the main theme of the book.

This reviewer would have been happier if the chapter on ergodicity were to cover a few more things that would connect it with the classical ergodic theory, though they are not needed in the subsequent chapters. However, this is a minor point. The author should be thanked for the book, the first of its kind, which would go a long way in generating interest in the subject.

Reviewer: S.G.Dani

##### MSC:

37A25 | Ergodicity, mixing, rates of mixing |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

22E40 | Discrete subgroups of Lie groups |

22F10 | Measurable group actions |

43A05 | Measures on groups and semigroups, etc. |