#### 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
##### 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