Word measures on GLn(q) and free group algebras

Ernst-West, Danielle; Puder, Doron; Seidel, Matan

doi:10.2140/ant.2024.18.2047

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Word measures on $\mathrm{GL}_{n}(q)$ and free group algebras

Danielle Ernst-West, Doron Puder and Matan Seidel

Vol. 18 (2024), No. 11, 2047–2090

DOI: 10.2140/ant.2024.18.2047

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 ${GL}_{N} (K)$ is obtained by sampling $r$ independent uniformly random elements $g_{1}, \dots, g_{r} \in {GL}_{N} (K)$ and evaluating $w (g_{1}, \dots, g_{r})$ . 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 $\lim_{N \to \infty} 𝔼_{w} [fix]$ 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, $𝔼_{w} [fix] = 2 + \frac{C}{q^{N}} + O (\frac{1}{q^{2 N}})$ , where $C$ is the number of rank-2 right ideals $I \leq K [F]$ 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 $I \leq K [F]$ 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.

Keywords

free group algebra, word measures, $q$-primitivity rank

Mathematical Subject Classification

Primary: 20C07, 20E05

Secondary: 16S34, 20C33, 20G40, 20H30, 68R15

Milestones

Received: 6 December 2022

Revised: 21 April 2023

Accepted: 27 November 2023

Published: 18 October 2024

Authors

Danielle Ernst-West
School of Mathematical Sciences Tel Aviv University Tel Aviv Israel

Doron Puder
School of Mathematical Sciences Tel Aviv University Tel Aviv Israel

Matan Seidel
School of Mathematical Sciences Tel Aviv University Tel Aviv Israel

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 18, No. 11, 2024