Fibers of word maps and the multiplicities of non-abelian composition factors

Abstract

We call a reduced word [Formula: see text] multiplicity-bounding if and only if a finite group on which the word map of [Formula: see text] has a fiber of positive proportion [Formula: see text] can only contain each non-abelian finite simple group [Formula: see text] as a composition factor with multiplicity bounded in terms of [Formula: see text] and [Formula: see text]. In this paper, based on recent work of Nikolov, we present methods to show that a given reduced word is multiplicity-bounding and apply them to give some nontrivial examples of multiplicity-bounding words, such as words of the form [Formula: see text], where [Formula: see text] is a single variable and [Formula: see text] an odd integer.