UCLA

We examine the distributions of non-commutative polynomials of non-atomic, freely independent random variables. In particular, we obtain an analogue of the Strong Atiyah Conjecture for free groups, thus proving that the measure of each atom of any n × n matricial polynomial of nonatomic, freely independent random variables is an integer multiple of n−1. In addition, we show that the Cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic, and thus the polynomial has a distribution that is real-analytic except at a finite number of points.