Kazhdan's Property (T)

This comprehensive introduction explores Property (T) and its applications across various fields in mathematics, including ergodic theory and operator algebras.

This book provides a thorough introduction to the concept of Property (T), a rigidity property for topological groups that was first introduced by D. Kazhdan in the mid-1960s. The primary goal of this property is to demonstrate that a significant class of lattices is finitely generated. The importance of Property (T) extends far beyond its original context, as it has been shown to play a critical role in numerous areas of mathematics, including discrete subgroups of Lie groups, ergodic theory, random walks, operator algebras, combinatorics, and theoretical computer science.

Kazhdan's Property (T) delves into the two main perspectives on Property (T): one that employs a unitary group representation approach and another that focuses on the fixed point property for affine isometric actions. Through these frameworks, the authors explore a variety of essential examples and applications across different mathematical domains. This comprehensive monograph serves as both an introduction and a reference for those looking to understand the significance and implications of Property (T).

Additionally, the book features a detailed appendix that systematically outlines key aspects of group representation theory that are necessary for formulating and developing Property (T). This resource is invaluable for mathematicians and students alike, providing clarity and depth to a complex yet fascinating subject.