We present a proof of a long-standing open problem in group theory, namely whether the automorphism group of the free group on five generators denoted ${\rm Aut}({\mathbb F}_5)$ has property $({\rm T})$. The proof relies on representation theory and incorporates computer assistance through the use of semidefinite programming. It was later shown that the groups ${\rm Aut}({\mathbb F}_5)$ for $n > 5$ also possess property $({\rm T})$ using a similar approach. We also discuss an application of this result in the form of an algorithm for generating random elements of finite groups called the product replacement algorithm.
|
