Kazhdan's property (T)

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In mathematics, a topological group G has property (T) if the trivial representation is isolated (that is, an isolated point) in the topological space of unitary representations, taken with the Fell topology. Informally, this means that if G acts on a complex Hilbert space unitarily (i.e. by isometries) and it has "almost invariant vectors", then it has an actual invariant vector. The definition gives this a precise, quantitative sense; it was introduced by David Kazhdan.

Groups with property (T) lead to good mixing properties: 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 matrices which can efficiently approximate any given 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.

1 Definitions
2 Examples
3 Applications
4 References

[edit] Definitions

First a little terminology is in place.

Let $\pi:G\rightarrow\operatorname{U}(H)$ be a unitary representation of a topological group G. If $ε > 0$ and $K$ is a compact subset of G, then a vector $ξ$

is called an $(ε, K)$ -invariant vector if $| | ξ - π(g)ξ | | < ε$ for all g in K.

There are several other, but equivalent, definitions for property (T) in use, such as the ones listed below.

Let G be a topological group. If every unitary representation of G that has, for all $ε > 0$ and compact $K\subset G$ , $(ε, K)$ -invariant vectors, has an actual invariant vector, then G has property (T).

Let G be a topological group. If there exists an $ε > 0$ and a compact subset K of G such that every unitary representation that has a $(ε, K)$ -invariant vector has an invariant vector, then G has property (T)

[edit] Examples

Lie groups with rank > 1 have property (T).
If G is a Lie group, and L is a lattice in G, then G has property (T) if and only if L does. So, for instance, the special linear group $S L (n, Z)$ has property (T), for n at least 3.
Quaternionic hyperbolic reflection groups have property (T); thus, some groups can be both hyperbolic, and have property (T).
An amenable group which has property (T) is compact. Amenability and property (T) are in a rough sense opposite: they make almost invariant vectors easy or hard to find.

[edit] Applications

Margulis used the fact that $S L (n, Z)$ 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 let 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 II factors with fundamental group smaller than the positive reals.

[edit] References

A. Lubotzky, Discrete groups, expanding graphs and invariant measures. Progress in Mathematics, 125. Birkhäuser Verlag, Basel, 1994. ISBN 3-7643-5075-X
[1] B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), unpublished.
[2] A. Lubotzky and A. Zuk, On property (τ), monograph to appear.
[3] A. Lubotzky, What is property (τ), AMS Notices 52 (2005), no. 6, 626-627.