property (T)
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| property (T) | |
|---|---|
| Name | property (T) |
| Introduced | 1967 |
| Introduced by | David Kazhdan |
| Area | Representation theory; Geometric group theory; Operator algebras |
| Key examples | SL(3,ℤ), Sp(n,1), E8 |
| Counterexamples | Free groups, Amenable groups, Infinite abelian groups |
| Related concepts | Kazhdan constant; Fixed point property; Superrigidity; Expanders |
property (T) is a rigidity property for topological groups, originally defined for locally compact second countable groups, that imposes strong restrictions on unitary representations and dynamical actions. It asserts that almost invariant vectors in unitary representations imply the existence of nonzero invariant vectors, producing fixed-point phenomena for actions on Hilbert spaces and compact spaces. The property has deep connections to representation theory, ergodic theory, operator algebras, and combinatorial constructions such as expander graphs.
Definition and equivalent formulations
A locally compact group G has property (T) if every continuous unitary representation of G on a Hilbert space that admits a net of almost invariant unit vectors actually contains a nonzero invariant vector. Equivalently, for compactly generated G there exists a finite subset S and ε>0 such that any unitary representation with an (S,ε)-almost invariant unit vector has a nonzero invariant vector; the optimal ε for a generating set S is the Kazhdan constant for (G,S). Alternate formulations appear via fixed-point properties: G has property (T) iff every continuous affine isometric action of G on a Hilbert space has a global fixed point, and via cohomology: vanishing of the first reduced cohomology H̄^1(G,π) for every unitary representation π. These formulations connect to work of David Kazhdan, Benoist, Delorme, Guichardet, and Zimmer.
Examples and non-examples
Classical examples include higher rank arithmetic groups and Lie groups: SL(n,ℝ) for n≥3, SL(n,ℤ) for n≥3, groups of type E8, and most higher rank simple algebraic groups over local fields, as studied by Kazhdan, Margulis, and Burger; many lattices in Sp(n,1) or in higher rank semisimple groups also have property (T). Examples also arise among compact groups and finite groups trivially. Non-examples include amenable groups such as infinite solvable groups, infinite abelian groups like ℤ, and free groups such as the free group F2; many mapping class groups and outer automorphism groups like Out(F_n) fail to have property (T), though special subgroups may. Certain products and wreath products can produce counterexamples studied by Shalom and Bekka.
Kazhdan constants and quantitative aspects
The Kazhdan constant gives a quantitative measure of how strongly a group exhibits property (T): for a finite generating set S of G, the Kazhdan constant κ(G,S) is the largest ε>0 witnessing the (S,ε)-criterion. Effective estimates of κ have been obtained for families such as SL(n,ℤ), congruence subgroups of SL_n(ℤ), and groups over local fields by work of Zuk, Lubotzky, Margulis, and Bourgain. Quantitative aspects influence spectral gaps for unitary representations, expanders constructed from Cayley graphs of finite quotients (as in the Lubotzky–Phillips–Sarnak theory), and norms of averaging operators studied in harmonic analysis on p-adic groups and Lie groups. Lower bounds for Kazhdan constants often use combinatorial or geometric estimates, while upper bounds relate to explicit almost invariant vectors coming from quasi-regular representations.
Rigidity, consequences, and applications
Property (T) implies several strong rigidity phenomena: lattices with property (T) satisfy forms of superrigidity and arithmeticity proved by Margulis; ergodic actions have spectral gaps used in equidistribution results and in the study of von Neumann algebras such as factors arising from group actions examined by Connes and Popa. Applications include constructions of families of expander graphs via finite quotients of property (T) groups used in computer science and combinatorics, fast mixing in random walks on groups studied by Diaconis, and rigidity in measured group theory impacting orbit equivalence and cost, investigated by Gaboriau and Furman. In operator algebras, property (T) yields strong constraints on deformation/rigidity phenomena, affecting classification of II1 factors and solidity properties studied by Ozawa and Ioana.
Stability, permanence properties, and inheritance
Property (T) is preserved under several group-theoretic operations but not all. It is preserved by taking finite extensions, passing to quotients, and taking products of groups with property (T); lattices in ambient groups with property (T) inherit property (T) under appropriate hypotheses as shown by Margulis and Kazhdan. However, property (T) is not preserved under passing to infinite index subgroups in general, and free products typically destroy property (T). Relative versions—Kazhdan pairs and property (T) for pairs of groups—give finer control for extensions and semi-direct products studied by Shalom and Bekka–de la Harpe–Valette.
Historical development and key results
Property (T) was introduced by David Kazhdan in 1967 in the context of representations of Lie groups and discrete subgroups. Early milestones include Margulis's use of property (T) in proving superrigidity and arithmeticity for higher rank lattices, the discovery of its connections to expanders by Margulis and later explicit constructions by Lubotzky, Phillips, and Sarnak, and the development of quantitative Kazhdan constants by Zuk and Burger. Subsequent advances linked property (T) to measured group theory, von Neumann algebra rigidity, and geometric group theory through contributions by Furman, Popa, Gaboriau, Shalom, Bekka, de la Harpe, and Valette. Contemporary research continues to explore refinements such as property (τ), relative property (T), and interactions with coarse geometry and expanders studied by Arzhantseva and Khukhro.