Property 'tau' (which is related to Kazhdan Property T in representation theory) has been found to have surprising applications to several areas of mathematics and computer science. We will introduce and review this property and discuss some of these applications, including to:
(1) costructions of expanders.
(2) producing efficient methods of generating random elements in groups;
(3) Thurston's conjecture on finite covers of hyperbolic manifolds.
We will present some open problems for topics for further research.
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