---
tag: GTFZ277047
kind: concept
name: Property (T)
msc: [22D55, 20F65]
sources:
  - url: https://en.wikipedia.org/wiki/Kazhdan's_property_(T)
same_as:
  wikidata: Q30765715
  wikipedia: Kazhdan's property (T)
---

## Summary

A locally compact group has property (T), or Kazhdan's property (T), under the following condition.
Every unitary representation that almost has invariant vectors actually has a nonzero invariant vector.
Kazhdan introduced it to show that lattices in groups such as $\mathrm{SL}_n(\R)$ with $n \ge 3$ are finitely generated.
It is a strong rigidity property, opposite to [[amenability|CEAT9BESKK]]: a locally compact group that is both amenable and has property (T) is compact.
It has applications to expander graphs, to ergodic theory and to the theory of operator algebras.

## Explanations

### L1 human

Let a group act by rotations, or more generally by unitary maps, on a space of possibly infinite dimension.
Suppose some unit vector is moved only a tiny bit by each element of a fixed generating set.
For most groups this tells you nothing, but for groups with property (T) it forces the existence of a vector that is not moved at all.
The integers fail this: shifting a long flat block of ones in $\ell^2(\Z)$ moves it very little, yet no nonzero vector of $\ell^2(\Z)$ is fixed by all shifts.
The groups $\mathrm{SL}_n(\Z)$ with $n \ge 3$ have property (T).