What is...?

Property (T) was defined by Kazhdan in the 1960s, who used it to prove two conjectures of Selberg on lattices in high-rank Lie groups. Shortly after that, Margulis used it to construct expander graphs.

Property $\tau$ is a baby version of property (T), but can be stated in several different equivalent ways and hence is useful for applications in several directions, some of them very unexpected. 

We will recall these properties and survey connections with hyperbolic manifolds,  3-manifolds, the congruence subgroup problem, super approximation, non-commutative dieve methods, and more. 

The talk is aimed at a non-expert audience.