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A recent paper on subfactors of von Neumann factors has stimulated
much research in von Neumann algebras. It was discovered soon after
the appearance of this paper that certain algebras which are used
there for the analysis of subfactors could also be used to define a
new polynomial invariant for links. Recent efforts to understand
the fundamental nature of the new link invariants has led to
connections with invariant theory, statistical mechanics and
quantum theory. In turn, the link invariants, the notion of a
quantum group, and the quantum Yang-Baxter equation have had a
great impact on the study of subfactors. Our subject is certain
algebraic and von Neumann algebraic topics closely related to the
original paper. However, in order to promote, in a modest way, the
contact between diverse fields of mathematics, we have tried to
make this work accessible to the broadest audience. Consequently,
this book contains much elementary expository material.
Groups as abstract structures were first recognized by
mathematicians in the nineteenth century. Groups are, of course,
sets given with appropriate "multiplications," and they are often
given together with actions on interesting geometric objects. But
groups are also interesting geometric objects by themselves. More
precisely, a finitely-generated group can be seen as a metric
space, the distance between two points being defined "up to
quasi-isometry" by some "word length," and this gives rise to a
very fruitful approach to group theory.
In this book, Pierre de la Harpe provides a concise and engaging
introduction to this approach, a new method for studying infinite
groups via their intrinsic geometry that has played a major role in
mathematics over the past two decades. A recognized expert in the
field, de la Harpe uses a hands-on presentation style, illustrating
key concepts of geometric group theory with numerous concrete
examples.
The first five chapters present basic combinatorial and geometric
group theory in a unique way, with an emphasis on
finitely-generated versus finitely-presented groups. In the final
three chapters, de la Harpe discusses new material on the growth of
groups, including a detailed treatment of the "Grigorchuk group,"
an infinite finitely-generated torsion group of intermediate growth
which is becoming more and more important in group theory. Most
sections are followed by exercises and a list of problems and
complements, enhancing the book's value for students; problems
range from slightly more difficult exercises to open research
questions in the field. An extensive list of references directs
readers to more advanced results as well as connectionswith other
subjects.
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).
Groups as abstract structures were first recognized by
mathematicians in the nineteenth century. Groups are, of course,
sets given with appropriate "multiplications," and they are often
given together with actions on interesting geometric objects. But
groups are also interesting geometric objects by themselves. More
precisely, a finitely-generated group can be seen as a metric
space, the distance between two points being defined "up to
quasi-isometry" by some "word length," and this gives rise to a
very fruitful approach to group theory.
In this book, Pierre de la Harpe provides a concise and engaging
introduction to this approach, a new method for studying infinite
groups via their intrinsic geometry that has played a major role in
mathematics over the past two decades. A recognized expert in the
field, de la Harpe uses a hands-on presentation style, illustrating
key concepts of geometric group theory with numerous concrete
examples.
The first five chapters present basic combinatorial and geometric
group theory in a unique way, with an emphasis on
finitely-generated versus finitely-presented groups. In the final
three chapters, de la Harpe discusses new material on the growth of
groups, including a detailed treatment of the "Grigorchuk group,"
an infinite finitely-generated torsion group of intermediate growth
which is becoming more and more important in group theory. Most
sections are followed by exercises and a list of problems and
complements, enhancing the book's value for students; problems
range from slightly more difficult exercises to open research
questions in the field. An extensive list of references directs
readers to more advanced results as well as connectionswith other
subjects.
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