Kazhdan's Property (T) - Discrete Groups

Discrete Groups

Historically property (T) was established for discrete groups 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 = SL_n(R) with R a ring and n≥ 3; the method relies on the fact that 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 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−1 h lies in S. If this graph is connected and the smallest non-zero eigenvalue of its Laplacian is greater than ½, then has property (T). A more general geometric version, due to Zuk and Ballmann & Swiatkowski (1997), states that if a discrete group 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 has property (T). Many new examples of hyperbolic groups with property (T) can be exhibited using this method.