Word measures on GLn(q) and free group algebras

Abstract

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 ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_N(K)$ is obtained by sampling $r$ independent uniformly random elements $g_1,\dots <!--FUNCTION\; APPLICATION-->,g_r\in {\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_N(K)$ and evaluating $w(g_1,\dots <!--FUNCTION\; APPLICATION-->,g_r)$. Consider ${\mathbb{𝔼}}_w[\mathrm{fix}<!--FUNCTION\; APPLICATION-->]$, the average number of vectors in $K^N$ fixed by a $w$-random element. We show that ${\mathbb{𝔼}}_w[\mathrm{fix}<!--FUNCTION\; APPLICATION-->]$ is a rational function in $q^N$. If $w=u^d$ with $u$ a nonpower, then the limit $\underset{N\to \infty }{\lim <!--FUNCTION\; APPLICATION-->}{\mathbb{𝔼}}_w[\mathrm{fix}<!--FUNCTION\; APPLICATION-->]$ depends only on $d$ and not on $u$. These two phenomena generalize to all stable characters of the groups ${\{{\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_N(K)\}}_N$.

A main feature of this work is the connection we establish between word measures on ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_N(K)$ 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, ${\mathbb{𝔼}}_w[\mathrm{fix}<!--FUNCTION\; APPLICATION-->]=2+\frac{C}{q^N}+O\left(\frac{1}{q^{2N}}\right)$, where $C$ is the number of rank-2 right ideals $I\le K[F]$ which contain $w-1$ 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 $I\le K[F]$ 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.