Surjectivity and equidistribution of the word x^ay^b on PSL(2,q) and SL(2,q)

TL;DR

This paper characterizes when the word map x^a y^b is surjective and almost equidistributed on PSL(2,q) and SL(2,q), using trace map analysis of positive words.

Contribution

It provides a complete classification of surjectivity conditions and demonstrates almost equidistribution for the word map on these groups.

Findings

Identifies conditions for surjectivity of x^a y^b on PSL(2,q) and SL(2,q)

Shows the word map is almost equidistributed for these groups

Uses trace map analysis of positive words in the proof

Abstract

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.