Kazhdan's property (T)

In

Let G be a σ-compact, locally compact topological group and π : G → U(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

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

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

(2) Any sequence of

(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

• Property (T) is preserved under quotients: if G has property (T) and H is a quotient group of G then H has property (T). Equivalently, if a homomorphic image of a group G does not have property (T) then G itself does not have property (T).
• If G has property (T) then G/[G, G] is compact.
• Any countable discrete group with property (T) is finitely generated.
• An amenable group which has property (T) is necessarily compact. Amenability and property (T) are in a rough sense opposite: they make almost invariant vectors easy or hard to find.
• Kazhdan's theorem: If Γ is a lattice in a Lie group G then Γ has property (T) if and only if G has property (T). Thus for n ≥ 3, the special linear group SL(n, Z) has property (T).

Examples

• Compact topological groups have property (T). In particular, the circle group, the additive group Z_p of p-adic integers, compact special unitary groups
  SU(n) and all finite groups have property (T).
• rank at least two have property (T). This family of groups includes the special linear groups SL(n, R) for n ≥ 3 and the special orthogonal groups SO(p,q) for p > q ≥ 2 and SO(p,p) for p ≥ 3. More generally, this holds for simple algebraic groups of rank at least two over a local field
  .
• The pairs (Rⁿ ⋊ SL(n, R), Rⁿ) and (Zⁿ ⋊ SL(n, Z), Zⁿ) have relative property (T) for n ≥ 2.
• For n ≥ 2, the noncompact Lie group Sp(n, 1) of isometries of a
  hermitian form of signature (n,1) is a simple Lie group of real rank 1 that has property (T). By Kazhdan's theorem, lattices in this group have property (T). This construction is significant because these lattices are hyperbolic groups; thus, there are groups that are hyperbolic and have property (T). Explicit examples of groups in this category are provided by arithmetic lattices in Sp(n, 1) and certain quaternionic reflection groups
  .

• The additive groups of integers Z, of real numbers R and of p-adic numbers Q_p.
• The special linear groups SL(2, Z) and SL(2, R), as a result of the existence of complementary series representations near the trivial representation, although SL(2,Z) has property (τ) with respect to principal congruence subgroups, by Selberg's theorem.
• Noncompact solvable groups.
• Nontrivial free groups and free abelian groups.

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.

• The algebraic method of Shalom applies when Γ = SL(n, R) with R a ring and n ≥ 3; the method relies on the fact that Γ can be boundedly generated, i.e. can be expressed as a finite product of easier subgroups, such as the elementary subgroups consisting of matrices differing from the identity matrix in one given off-diagonal position.
• The geometric method has its origins in ideas of Garland,
  link at each vertex, then Γ has property (T). Many new examples of hyperbolic groups
  with property (T) can be exhibited using this method.
• The computer-assisted method is based on a suggestion by Narutaka Ozawa and has been successfully implemented by several researchers. It is based on the algebraic characterization of property (T) in terms of an inequality in the real group algebra, for which a solution may be found by solving a semidefinite programming problem numerically on a computer. Notably, this method has confirmed property (T) for the automorphism group of the free group of rank at least 5. No human proof is known for this result.

Applications

• expanding graphs
  , that is, graphs with the property that every subset has a uniformly large "boundary". This connection led to a number of recent studies giving an explicit estimate of Kazhdan constants, quantifying property (T) for a particular group and a generating set.
• positive reals
  ${\displaystyle \mathbb {R} ^{+}}$. Sorin Popa subsequently used relative property (T) for discrete groups to produce a type II₁ factor with trivial fundamental group.
• Groups with property (T) also have Serre's property FA.^[1]
• Toshikazu Sunada observed that the positivity of the bottom of the spectrum of a "twisted" Laplacian on a closed manifold is related to property (T) of the fundamental group.^[2] This observation yields Brooks' result which says that the bottom of the spectrum of the Laplacian on the universal covering manifold over a closed Riemannian manifold M equals zero if and only if the fundamental group of M is amenable.^[3]

References