Word Measures on $G{L}_N(q)$ 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 $G{L}_N(K)$ is obtained by sampling $r$ independent uniformly random elements $g_1,\dots ,g_r\in G{L}_N(K)$ and evaluating $w\left(g_1,\dots ,g_r\right)$. Consider ${\mathbb{E}}_w\left[\mathrm{fix}\right]$, the average number of vectors in $K^N$ fixed by a $w$-random element. We show that ${\mathbb{E}}_w\left[\mathrm{fix}\right]$ is a rational function in $q^N$. Moreover, if $w=u^d$ with $u$ a non-power, then the limit ${\lim }_{N\to \infty }{\mathbb{E}}_w\left[\mathrm{fix}\right]$ depends only on $d$ and not on $u$. These two phenomena generalize to all stable characters of the groups ${\left\{G{L}_N(K)\right\}}_N$. A main feature of this work is the connection we establish between word measures on $G{L}_N(K)$ and the free group algebra $K\left[F\right]$. A classical result of Cohn [1964] and Lewin [1969] is that every one-sided ideal of $K\left[F\right]$ is a free $K\left[F\right]$-module with a well-defined rank. We show that for $w$ a non-power, ${\mathbb{E}}_w\left[\mathrm{fix}\right]=2+\frac{C}{q^N}+O\left(\frac{1}{q^{2N}}\right)$, where $C$ is the number of rank-2 right ideals $I\le K\left[F\right]$ 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 $I\le K\left[F\right]$ is a right ideal with a generating set supported on $T$, then $I$ admits a basis supported on $T$. We also prove an analogue of Kaplansky's unit conjecture for certain $K\left[F\right]$-modules.

arXiv comment39 pages, 2 figures; Journal version, after a minor revision and with updated references

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