Measures induced by words on GL_n(q) and free group algebras

Fix a finite field K and a word w in a free group F.  A  w-random element in GL_n(K) is obtained by substituting the letters of w with uniform random elements of GL_n(K). For example, if w = abab^{-2}, a w-random element is ghgh^{-2} with g,h independent and uniformly random in GL_n(K). The moments of w-random elements reveal a surprising structure which relates to the free group algebra K[F].

In this talk I will describe what we know about this structure, and draw some analogies to w-random permutations.

This is joint work with Doron Puder and Matan Seidel.