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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).
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.
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.
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