Kazhdan's Property (T) - Examples

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).
• Simple real Lie groups of real 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 quaternionic 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.

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

• The additive groups of integers Z, of real numbers R and of p-adic numbers Q_p.
• The special linear groups SL₂(Z) and SL₂(R), although SL₂ has property (τ) with respect to principal congruence subgroups, by Selberg's theorem.
• Noncompact solvable groups.
• Nontrivial free groups and free abelian groups.