Most words are geometrically almost uniform

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 $\left|G\right|\to \infty$. In this paper, we show that the proportion is in fact $1$.