Fourier and Small ball estimates for word maps on unitary groups

Abstract: Let $w(x,y)$ be a word in a free group. For any group $G$, $w$ induces a word map $w:G^2 \to G$. For example, the commutator word $w=xyx^{-1}y^{-1}$ induces the commutator map. If $G$ is finite, one can ask what is the probability that $w(g,h)$ is equal to the identity element $e$, for a pair $(g,h)$ of elements in $G$, chosen independently at random. In the setting of finite simple groups, Larsen and Shalev showed there exists $\epsilon(w)>0$ (depending only on $w$), such that the probability that $w(g,h)=e$ is smaller than $|G|^{-\epsilon(w)}$, whenever $G$ is large enough (depending on $w$). In this talk, I will discuss analogous questions for compact groups, with a focus on the family of unitary groups; For example, given a word $w$, and given two independent Haar-random $n$ by $n$ unitary matrices $A$ and $B$, what is the probability that $w(A,B)$ is contained in a small ball around the identity matrix?

Based on a joint work with Nir Avni and Michael Larsen.

group theory

Audience: researchers in the topic

Groups in Galway 2024