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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
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 g1,,gr GL N(K) and evaluating w(g1,,gr). Consider 𝔼w[fix ], the average number of vectors in KN fixed by a w-random element. We show that 𝔼w[fix ] is a rational function in qN. If w = ud with u a nonpower, then the limit lim N𝔼w[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, 𝔼w[fix ] = 2 + C qN + O( 1 q2N), where C is the number of rank-2 right ideals I 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 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

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