Most words are geometrically almost uniform

Larsen, Michael Jeffrey

doi:10.2140/ant.2020.14.2185

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Most words are geometrically almost uniform

Michael Jeffrey Larsen

Vol. 14 (2020), No. 8, 2185–2196

DOI: 10.2140/ant.2020.14.2185

Abstract

If $w$ is a word in $d > 1$ letters and $G$ is a finite group, evaluation of $w$ on a uniformly randomly chosen $d$ -tuple in $G$ gives a random variable with values in $G$ , which may or may not be uniform. It is known that if $G$ ranges over finite simple groups of given root system and characteristic, a positive proportion of words $w$ give a distribution which approaches uniformity in the limit as $| G | \to \infty$ . In this paper, we show that the proportion is in fact $1$ .

Keywords

word maps, random walks on finite simple groups, groups of Lie type

Mathematical Subject Classification 2010

Primary: 20P05

Secondary: 11G25, 14G15, 20G40

Milestones

Received: 16 October 2019

Revised: 17 February 2020

Accepted: 25 March 2020

Published: 18 September 2020

Authors

Michael Jeffrey Larsen
Department of Mathematics Indiana University Rawles Hall Bloomington, IN United States

Vol. 14, No. 8, 2020