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.