Kazhdan's property (T)

In mathematics, a locally compact topological group G has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology. Informally, this means that if G acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero invariant vector. The formal definition, introduced by David Kazhdan (1967), gives this a precise, quantitative meaning.

Although originally defined in terms of irreducible representations, property (T) can often be checked even when there is little or no explicit knowledge of the unitary dual. Property (T) has important applications to group representation theory, lattices in algebraic groups over local fields, ergodic theory, geometric group theory, expanders, operator algebras and the theory of networks.

Definitions

Let G be a σ-compact, locally compact topological group and π : GU(H) a unitary representation of G on a (complex) Hilbert space H. If ε > 0 and K is a compact subset of G, then a unit vector ξ in H is called an (ε, K)-invariant vector if

 \forall g \in K \ : \ \left \|\pi(g) \xi - \xi \right \| < \varepsilon.

The following conditions on G are all equivalent to G having property (T) of Kazhdan, and any of them can be used as the definition of property (T).

(1) The trivial representation is an isolated point of the unitary dual of G with Fell topology.

(2) Any sequence of continuous positive definite functions on G converging to 1 uniformly on compact subsets, converges to 1 uniformly on G.

(3) Every unitary representation of G that has an (ε, K)-invariant unit vector for any ε > 0 and any compact subset K, has a non-zero invariant vector.

(4) There exists an ε > 0 and a compact subset K of G such that every unitary representation of G that has an (ε, K)-invariant unit vector, has a nonzero invariant vector.

(5) Every continuous affine isometric action of G on a real Hilbert space has a fixed point (property (FH)).

If H is a closed subgroup of G, the pair (G,H) is said to have relative property (T) of Margulis if there exists an ε > 0 and a compact subset K of G such that whenever a unitary representation of G has an (ε, K)-invariant unit vector, then it has a non-zero vector fixed by H.

Discussion

Definition (4) evidently implies definition (3). To show the converse, assume local compactness. So let G be a locally compact group satisfying (3). By Theorem 1.3.1 of Bekka et al., G is compactly generated. Therefore, Remark 1.1.2(v) of Bekka et al. implies the following: if we take K to be a compact generating set of G, and let ε be any positive real number, then a unitary representation of G having an (ε, K)-invariant unit vector has (ε', K ')-invariant unit vectors for every ε' > 0 and K ' compact. Therefore, by (3), such a representation of G will have a nonzero invariant vector, establishing (4).

The equivalence of (4) and (5) (Property (FH)) is the Delorme-Guichardet theorem. The fact that (5) implies (4) requires the assumption that G is σ-compact (and locally compact) (Bekka et al., Theorem 2.12.4).

General properties

Examples

Examples of groups that do not have property (T) include

Discrete groups

Historically property (T) was established for discrete groups Γ by embedding them as lattices in real or p-adic Lie groups with property (T). There are now several direct methods available.

Applications

References

  1. Watatani, Yasuo (1981). "Property T of Kazhdan implies property FA of Serre.". Math. Japon. 27: 97–103. MR MR649023. Zbl 0489.20022.
  2. Sunada, Toshikazu (1989). "Unitary representations of fundamental groups and the spectrum of twisted Laplacians". Topology 28: 125–132. doi:10.1016/0040-9383(89)90015-3.
  3. Brooks, Robert (1981). "The fundamental group and the spectrum of the Laplacian". Comment. Math. Helv. 56: 581–598. doi:10.1007/bf02566228.
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