Fibers of word maps and some applications

Abstract

Every word w in the free group Fd defines for each group G a word map, also denoted w, from Gd to G. We prove that for all w≠1 there exists ϵ>0 such that for all finite simple groups G and all g∈G,|w−1(g)|=O(|G|d−ϵ), where the implicit constant depends only on w. In particular the probability that w(g1,…,gd)=1 is at most |G|−ϵ for some ϵ>0 and all large finite simple groups G. This result is then applied in the context of subgroup growth and representation varieties.