On surjectivity of word maps on $\mathrm{PSL}_2$

TL;DR

This paper proves that certain double commutator word maps are surjective on the projective special linear group over algebraically closed fields of characteristic zero, expanding understanding of word map properties in algebraic groups.

Contribution

It establishes the surjectivity of a specific class of double commutator word maps on $ ext{PSL}_2(K)$ over algebraically closed fields of characteristic zero, a new result in algebraic group theory.

Findings

Double commutator words are surjective on $ ext{PSL}_2(K)$

Surjectivity holds over algebraically closed fields of characteristic zero

Advances understanding of word maps in algebraic groups

Abstract

Let $w = [[x^k, y^l], [x^m, y^n]]$ be a non-trivial double commutator word. We show that $w$ is surjective on $\mathrm{PSL}_2(K)$, where $K$ is an algebraically closed field of characteristic $0$.