Criteria for equidistribution of solutions of word equations on SL(2)

TL;DR

This paper establishes criteria based on trace polynomials for when solutions to word equations in SL(2,q) are equidistributed, identifying classes of words with this property and demonstrating its generic nature.

Contribution

It introduces explicit criteria for equidistribution of solutions in SL(2,q) based on trace polynomials and characterizes classes of words with this property.

Findings

Criteria for equidistribution derived from trace polynomials

Identification of classes of words with equidistribution property

Demonstration that equidistribution is generic within these classes

Abstract

We study equidistribution of solutions of word equations of the form w(x,y)=g in the family of finite groups SL(2,q). We provide criteria for equidistribution in terms of the trace polynomial of w. This allows us to get an explicit description of certain classes of words possessing the equidistribution property and show that this property is generic within these classes.