SURJECTIVITY AND EQUIDISTRIBUTION OF THE WORD xayb ON PSL(2, q) AND SL(2, q)

Abstract

We determine the integers a, b ≥ 1 and the prime powers q for which the word map w(x, y) = xayb is surjective on the group PSL (2, q) (and SL (2, q)). We moreover show that this map is almost equidistributed for the family of groups PSL (2, q) (and SL (2, q)). Our proof is based on the investigation of the trace map of positive words.