at least 5. No human proof is known for this result. Applications Grigory Margulis used the fact that SL(n, Z) (for n ≥ 3) has property (T) to construct explicit families of 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. Alain Connes used discrete groups with property (T) to find examples of type II1 factors with countable fundamental group, so in particular not the whole of positive reals 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) lead to good mixing properties in ergodic theory: again informally, a process which mixes slowly leaves some subsets almost invariant. Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can efficiently approximate any given invertible matrix, in the sense that every matrix can be approximated, to a high degree of accuracy, by a finite product of matrices in the list or their inverses, so that the number of matrices needed is proportional to the number of significant digits in the approximation. Groups with property (T) also have Serre's property FA. 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. 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. References . Lubotzky, A. and A. Zuk, On property (τ), monograph to appear. . . . Unitary representation theory Topological groups Geometric group theory Computer-assisted proofs