Vol. 14, No. 8, 2020

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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

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|. In this paper, we show that the proportion is in fact 1.

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word maps, random walks on finite simple groups, groups of Lie type
Mathematical Subject Classification 2010
Primary: 20P05
Secondary: 11G25, 14G15, 20G40
Received: 16 October 2019
Revised: 17 February 2020
Accepted: 25 March 2020
Published: 18 September 2020
Michael Jeffrey Larsen
Department of Mathematics
Indiana University
Rawles Hall
Bloomington, IN
United States