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**Kazhdan’s property.**
*(English)*
Zbl 1146.22009

New Mathematical Monographs 11. Cambridge: Cambridge University Press (ISBN 978-0-521-88720-5/hbk). xiii, 472 p. £ 50.00; $ 99.00; $ 59.99/e-book (2008).

A locally compact group \(G\) has Kazhdan’s property \((T)\) if its trivial one dimensional representation is isolated in \(\widehat{G}\), the unitary dual of \(G\) equipped with a natural topology. D. Kazhdan introduced this property 1967 in a short note [Funct. Anal. Appl. 1, 63–65 (1967; Zbl 0168.27602)]. He proved that this property is satisfied by all simple Lie groups of real rank \(\geq 2\) and is inherited by subgroup of finite covolume. Solving a conjecture of Siegel, Kazhdan deduced from this that a lattice \(\Gamma\) in such a Lie group is finitely generated and its commutator subgroup \(\Gamma'\) has finite index. This is quite spectacular and shows that property \((T)\) is an extremely powerful tool. It was shown in the late 1970s that for a large class of groups property \((T)\) is equivalent to property \((FH)\), where a topological group \(G\) is said to have property \((FH)\) if every continuous action of \(G\) by affine isometries on a Hilbert space has a fixed point. Later developments have shown that property \((T)\) plays an important role in an amazingly large variety of subjects, including discrete subgroups of Lie groups, ergodic theory, random walks, operator algebras, combinatorics, and theoretical computer sciences. The monograph under review offers a comprehensive introduction to the theory. It describes the two most important points of view on property \((T)\): the first uses a unitary representation approach, and the second a fixed point property for affine isometric actions. Via these the authors discuss a wide range of important examples and applications to several domains of mathematics. The book is divided into two parts. The chapters of Part I are the following:

(1) Definitions, first consequences, and basic examples. (2) Property \((FH)\). (3) Reduced cohomology. (4) Bounded generation. (5) A spectral criterion for property \((T)\). (6) Some applications of property \((T)\). (7) A short list of open questions.

Part II contains the background of unitary representations of locally compact groups. The themes of Part II are:

(A) Unitary group representations. (B) Measures on homogeneous spaces. (C) Functions of positive type and \(GNS\) constructions. (D) Unitary representations of locally compact abelian groups. (E) Induced representations. (F) Weak containment and Fell’s topology. (G) Amenability.

Also the book contains a historical introduction and a comprehensive bibliography. This well-written book will serve graduate and postgraduate students as well as researchers.

(1) Definitions, first consequences, and basic examples. (2) Property \((FH)\). (3) Reduced cohomology. (4) Bounded generation. (5) A spectral criterion for property \((T)\). (6) Some applications of property \((T)\). (7) A short list of open questions.

Part II contains the background of unitary representations of locally compact groups. The themes of Part II are:

(A) Unitary group representations. (B) Measures on homogeneous spaces. (C) Functions of positive type and \(GNS\) constructions. (D) Unitary representations of locally compact abelian groups. (E) Induced representations. (F) Weak containment and Fell’s topology. (G) Amenability.

Also the book contains a historical introduction and a comprehensive bibliography. This well-written book will serve graduate and postgraduate students as well as researchers.

Reviewer: Sergei Platonov (Petrozavodsk)

### MSC:

22D10 | Unitary representations of locally compact groups |

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

22E46 | Semisimple Lie groups and their representations |

22E40 | Discrete subgroups of Lie groups |

22E99 | Lie groups |

05C80 | Random graphs (graph-theoretic aspects) |

05C99 | Graph theory |

53C35 | Differential geometry of symmetric spaces |