Uniform words are primitive

Let $G$ be a finite group, and let $a$, $b$, $c$,... be independent random elements of $G$, chosen at uniform distribution. What is the distribution of the element obtained by a fixed word in the letters $a$, $b$, $c$,..., such as $ab$, $a^2$, or $aba^{-2}b^{-1}$? More concretely, do these new random elements have uniform distribution? In general, a word $w$ in the free group $F_k$ is called uniform if it induces the uniform distribution on every finite group $G$. So which words are uniform? A large set of uniform words are those which are 'primitive' in the free group $F_k$, namely those belonging to some basis (a free generating set) of $F_k$. Several mathematicians have conjectured that primitive words are the only uniform words. In a joint work with O. Parzanchevski, we prove this conjecture. I will try to define and explain all notions, and give many details from the proof. I will also present related open problems.