Kazhdan's property (T)

Kazhdan's property (T)
Name	Kazhdan's property (T)
Field	Mathematics
Introduced	1967
Introduced by	David Kazhdan

Kazhdan's property (T) is a rigidity property for topological groups introduced by David Kazhdan in 1967 that constrains unitary representations and forces spectral gaps for averaging operators. It plays a central role in the study of discrete groups such as SL_n(ℤ), lattices in Lie groups like SL_2(ℝ), and has deep connections to operator algebras, expander graphs, and ergodic theory. Property (T) links representation theory, combinatorial constructions, and geometric group theory through quantitative invariants such as Kazhdan constants and spectral gaps.

Definition and equivalent formulations

A locally compact group G has property (T) if every unitary representation of G on a Hilbert space that has almost invariant vectors contains a nonzero invariant vector; this definition appears in works by David Kazhdan and is equivalent to the existence of a compact subset Q and ε>0 (a Kazhdan pair) such that any unitary representation with a (Q,ε)-invariant unit vector has a nonzero invariant vector. Equivalent formulations involve the isolation of the trivial representation in the unitary dual of G, the presence of a spectral gap for the Laplacian on G when acting by the regular representation, and fixed point properties for affine isometric actions on Hilbert spaces, as used in studies by Michael Gromov and Bennett Farb.

Historical background and motivation

Property (T) originated in Kazhdan’s 1967 paper motivated by rigidity phenomena in lattices of semisimple Lie groups and representation-theoretic control of automorphic forms studied in the context of Selberg trace formula and Harish-Chandra theory. Subsequent development involved contributions from George Mackey, Israel Gelfand, Harish-Chandra on unitary duals, and work by Margulis who applied rigidity to superrigidity and arithmeticity of lattices in SL_n(ℝ), Sp_n(ℝ), and SO(n,1). Later interactions with combinatorics and computer science emerged via constructions of expander graphs by Lubotzky, Phillips, and Sarnak, linking property (T) to spectral expansion and connections to the Ramanujan graph program.

Examples and non-examples

Classical examples include higher-rank simple Lie groups such as SL_n(ℝ) for n≥3 and their lattices including SL_n(ℤ), as shown by Kazhdan and extended by Margulis. Compact groups and groups with property (T) like SU(n) occur in representation-theoretic contexts. Notable non-examples are abelian groups such as ℝ and ℤ, free groups like F_2, and hyperbolic groups such as many instances of Gromov random groups in certain density ranges; these fail property (T) because of abundant almost invariant vectors or nontrivial actions on trees studied by Jean-Pierre Serre. Some groups, for instance Sp(n,1) for n≥2, occupy a borderline status with respect to related properties like property (FA) and property (FH).

Basic properties and permanence

Property (T) is preserved under taking quotients and finite index extensions, and is inherited from lattices to ambient groups under suitable conditions as in Margulis’s superrigidity framework. It is not stable under free products without amalgamation, as free products of nontrivial groups typically fail property (T). For direct products, property (T) holds if and only if each factor has property (T). Permanence results exploit techniques from representation theory developed by Harish-Chandra and structural theory of semisimple Lie groups; conversely, induction and restriction of representations illustrate limitations exemplified in examples by Burger and Mozes.

Representations, spectral gap, and Kazhdan constants

In analytic terms, property (T) is equivalent to a uniform spectral gap for the averaging operator associated to a generating set or a compact subset, a viewpoint exploited by Lubotzky and Zuk in constructing expanders. The Kazhdan constant quantifies how large ε can be for a given compact Q and is a delicate invariant related to eigenvalue gaps of Laplacians on Cayley graphs of finitely generated groups; computations involve methods from operator algebra theory and harmonic analysis on Lie groups developed by Eymard and Connes. The presence of a spectral gap has strong consequences for mixing rates in unitary representations, ergodic theorems studied by Furstenberg and Zimmer, and quantitative equidistribution results in arithmetic contexts like the Erdős–Rényi random graph analogy.

Applications in geometry, group theory, and operator algebras

Property (T) underpins rigidity results in the theory of lattices and affects deformation spaces encountered in the work of Mostow and Margulis. It provides a mechanism to construct families of expander graphs used in computer science and combinatorics by Lubotzky, Phillips, and Sarnak. In operator algebras, property (T) influences structure and classification results for von Neumann algebras and C*-algebras, featuring in work by Connes, Jones, and Popa on rigidity and deformation/rigidity theory. Applications also appear in measurable group theory and orbit equivalence studied by Gaboriau and Furman, and in topological dynamics through fixed point properties examined by Shalom.

Variants and generalizations

Several generalizations of property (T) have been introduced: relative property (T) for pairs (G,H) used in Margulis’s superrigidity arguments; property (FH) concerning fixed points in Hilbert spaces linked to affine isometric actions by Delorme and Guichardet; property (τ) for families of finite index subgroups exploited by Lubotzky in expander constructions; and stronger notions like property (TT) or property (T) with respect to Banach spaces investigated by BFGM collaborators and Lafforgue who applied these to obstruct Baum–Connes type conjectures for certain groups. These variants connect to modern directions in higher rank rigidity, geometric group theory, and noncommutative geometry pursued by Gromov, Lafforgue, and Monod.

Category:Topological groups