Fibers of word maps and some applications

Every word w in the free group F _d defines for each group G a word map, also denoted w, from G ^d 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.