Some word maps that are non-surjective on infinitely many finite simple groups

TL;DR

This paper presents the first examples of specific words in a free group that are not proper powers and whose associated word maps fail to be surjective on infinitely many finite simple groups.

Contribution

It introduces the first known instances of such words with non-surjective maps across an infinite family of finite simple groups.

Findings

Identified non-surjective word maps on infinitely many finite simple groups.

Provided explicit examples of non-power words with this property.

Expanded understanding of word maps in finite group theory.

Abstract

We provide the first examples of words in the free group of rank 2 which are not proper powers and for which the corresponding word maps are non-surjective on an infinite family of finite non-abelian simple groups.