Fibers of word maps and some applications

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