Word maps with small image in simple groups

TL;DR

This paper constructs specific words in free groups that produce small or controlled images in certain finite simple groups, demonstrating non-surjectivity and precise element class images.

Contribution

It introduces new non-power words with small images in SL(2, 2^n) and Alt(n), expanding understanding of word maps in simple groups.

Findings

Constructed non-power words with small images in SL(2, 2^n)

Built words with images limited to identity and a single element class in Alt(n)

Demonstrated non-surjectivity of certain word maps in these groups

Abstract

We construct non-power words which have small image in SL(2; 22n) for each n. In particular, the corresponding word maps are non-surjective. We also use this to construct word maps whose values are precisely the identity and a single equivalence class of elements of order 17. In the second part we construct words which have image consisting of the identity and a single equivalence class of elements in Alt(n) for all n for any equivalence class with support size at most 10.