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 Qp. 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, Gromov and Pierre Pansu. Its simplest combinatorial version is due to Zuk: let Γ be a discrete group generated by a finite subset S, closed under taking inverses and not containing the identity, and define a finite graph with vertices S and an edge between g and h whenever g−1h lies in S. If this graph is connected and the smallest non-zero eigenvalue of the Laplacian of the corresponding simple random walk is greater than ½, then Γ has property (T). A more general geometric version, due to Zuk and , states that if a discrete group Γ acts properly discontinuously and cocompactly on a contractible 2-dimensional simplicial complex with the same graph theoretic conditions placed on the 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