Surjectivity and equidistribution of the word x^ay^b on PSL(2,q) and   SL(2,q)
  We determine the positive integers a,b and the prime powers q for which the word map w(x,y)=x^ay^b 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. 
Tatiana Bandman and Shelly Garion
math.GR math.AG