Event № 173
We study measures induced by free words on the unitary groups U (n): let w be a word in the free group F_r on r generators x_1,...,x_r. For every i=1,...,r substitute x_i with an independent, Haar-distributed random element of U(n) and evaluate the product defined by w to obtain a random element in U(n). The measure of this element is called the w-measure on U(n).Let Tr_w(n) denote the expected trace of a random unitary matrix sampled from U (n) according to the w-measure. It was shown by Voiculescu (91') that for w \ne 1, this expected trace is o(n) asymptotically in n. We relate the numbers Tr_w(n) to the theory of commutator length of words and obtain a much stronger statement. Our analysis also sheds new light on the solutions of the equation [u_1, v_1] . . . [u_g, v_g] = w in free groups. I will also present some interesting related open problems.
Joint work with Michael Magee.