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This question is inspired by Probability of commutation in a compact group, which asked whether $P(xyx^-1y^-1 = 1)$ could take values strictly between $0$ and $1$ on a compact connected group. That question turned out to have a rather easy answer, but one that was quite particular to the word $xyx^-1y^-1$.

Question: Let $w$ be an arbitrary word in $k$ letters, and let $G$ be a compact connected group. Let $x_1, dots, x_k$ be drawn uniformly from $G$. Can $P(w(x_1, dots, x_k)=1)$ be strictly between $0$ and $1$?

By the Peter--Weyl theorem we may assume that $G$ is a compact connected Lie group.

There is some low-hanging fruit, like $w = [[x,y],z]$, but that's still very specialized. Admittedly I am not even clear on the words $w = x^n$.

The negative answer follows easily from a very useful fact that should be better known, so I am writing it explicitly.

Fact (Tannaka, Chevalley):
Every (connected) compact Lie group is isomorphic as a topological group to the group of real points of a (connected) reductive affine real algebraic group.
Moreover, the Haar measure on such a group is given by a volume form on the corrseponding variety.

We thus may view (for a given word $w$ in the rank $k$ free group $F_k$) the word map $w:G^kto G$ as a morphism of real varieties and its solution space (the fiber over the identity element) as a the real points of a closed real subvariety. Since $G$ is connected, this subvariety will be either $G^k$ itself, or a lower dimensional one. Its measure, accordingly, will be either 0 or 1.

We could be more precise:

• If $G$ is trivial then the measure is 1.




• If $G$ is non-trivial abelian then it is isomprphic to a torus $textSO(2)^n$, and it is easy to see that the measure is 1 iff $w$ is in the commutator group of $F_k$, otherwise the measure is 0.




• If $G$ is non-abelian then it contains a free group on $k$ generators (by Tits alternative, if you want to hit it with a hammer), thus the corresponding subvariety is proper and its measure is necessarily 0.