Word measures on GL_N (q) and free group algebras

Fix a finite field K of order q and a word w in a free group F on r generators. A w-random element in GL_N (K) is obtained by sampling r independent uniformly random elements g₁, …, g_r ∈ GL_N (K) and evaluating w(g₁, …, g_r). Consider E_w [fix], the average number of vectors in K^N fixed by a w-random element. We show that E_w [fix] is a rational function in q^N. If w = u^d with u a nonpower, then the limit lim_N→∞ E_w [fix] depends only on d and not on u. These two phenomena generalize to all stable characters of the groups {GL_N (K)}_N. A main feature of this work is the connection we establish between word measures on GL_N (K) and the free group algebra K [F]. A classical result of Cohn (1964) and Lewin (1969) is that every one-sided ideal of K [F] is a free K [F]-module with a well-defined rank. We show that for w a nonpower, (Formula presented), where C is the number of rank-2 right ideals I ≤ K [F] which contain w − 1^2N but not as a basis element. We describe a full conjectural picture generalizing this result, featuring a new invariant we call the q-primitivity rank of w. In the process, we prove several new results about free group algebras. For example, we show that if T is any finite subtree of the Cayley graph of F, and I ≤ K [F] is a right ideal with a generating set supported on T, then I admits a basis supported on T. We also prove an analog of Kaplansky’s unit conjecture for certain K [ F]-modules.