Non-commutative optimization and Kazhdan property (T)

During the talk I will describe a striking application of the non-commutative optimisation to a problem in geometric group theory, namely the Kazhdan property (T).

It is known that property (T) is equivalent to positivity of the element $\Delta^2 - \lambda\Delta$ in the full group $*$-algebra, where $\Delta$ is group Laplacian associated to a generating set. It turns out that its positivity is equivalent to the existence of a sum of (hermitian) squares decomposition of $\Delta^2 - \lambda\Delta$ in the real group algebra, which might be understood as an algebra in transition between the classical polynomial algebra and Heltons free algebra.

I will mention the algorithm encoding the optimisation problem, and how an (imprecise) numerical solution can be turned into a mathematical proof. We applied the method to answer the question of property (T) for $\operatorname{Aut}(F_{n})$, the automorphism group of the free group on $n$ generators. Since (even for $n=4$) the size of the problem is out of reach of the method, I will show how to exploit the inherent symmetry of the problem to obtain a smaller, equivalent problem. This leads to constructive, computer-assisted proof that $\operatorname{Aut}(F_{n})$ has Kazhdan\'s property (T).

Property (T) for $\operatorname{Aut}(F_{n})$ has been a long-standing open problem and, as observed by Lubotzky and Pak, the positive resolution leads to better understanding of the effectiveness of the product replacement algorithm commonly used in computational group theory.