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Unitary representations of groups play an important role in many
subjects, including number theory, geometry, probability theory,
partial differential equations, and quantum mechanics. This
monograph focuses on dual spaces associated to a group, which are
spaces of building blocks of general unitary representations.
Special attention is paid to discrete groups for which the unitary
dual, the most common dual space, has proven to be not useful in
general and for which other duals spaces have to be considered,
such as the primitive dual, the normal quasi-dual, or spaces of
characters. The book offers a detailed exposition of these
alternative dual spaces and covers the basic facts about unitary
representations and operator algebras needed for their study.
Complete and elementary proofs are provided for most of the
fundamental results that up to now have been accessible only in
original papers and appear here for the first time in textbook
form. A special feature of this monograph is that the theory is
systematically illustrated by a family of examples of discrete
groups for which the various dual spaces are discussed in great
detail: infinite dihedral group, Heisenberg groups, affine groups
of fields, solvable Baumslag-Solitar group, lamplighter group, and
general and special linear groups. The book will appeal to graduate
students who wish to learn the basics facts of an important topic
and provides a useful resource for researchers from a variety of
areas. The only prerequisites are a basic background in group
theory, measure theory, and operator algebras.
Property (T) is a rigidity property for topological groups, first
formulated by D. Kazhdan in the mid 1960's with the aim of
demonstrating that a large class of lattices are finitely
generated. 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 science.
This monograph offers a comprehensive introduction to the theory.
It describes the two most important points of view on Property (T):
the first uses a unitary group representation approach, and the
second a fixed point property for affine isometric actions. Via
these the authors discuss a range of important examples and
applications to several domains of mathematics. A detailed appendix
provides a systematic exposition of parts of the theory of group
representations that are used to formulate and develop Property
(T).
The study of geodesic flows on homogenous spaces is an area of
research that has yielded some fascinating developments. This book,
first published in 2000, focuses on many of these, and one of its
highlights is an elementary and complete proof (due to Margulis and
Dani) of Oppenheim's conjecture. Also included here: an exposition
of Ratner's work on Raghunathan's conjectures; a complete proof of
the Howe-Moore vanishing theorem for general semisimple Lie groups;
a new treatment of Mautner's result on the geodesic flow of a
Riemannian symmetric space; Mozes' result about mixing of all
orders and the asymptotic distribution of lattice points in the
hyperbolic plane; Ledrappier's example of a mixing action which is
not a mixing of all orders. The treatment is as self-contained and
elementary as possible. It should appeal to graduate students and
researchers interested in dynamical systems, harmonic analysis,
differential geometry, Lie theory and number theory.
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