Year: 2023
Type: preprint
Abstract: We study the word image of words with constants in ${\rm GL}(V)$ and show that it is large provided the word satisfies some natural conditions on its length and its critical constants. There are various consequences: We prove that for every $l \geq 1$, there are only finitely many pairs $(n,q)$ such that the length of the shortest non-singular mixed identity ${\rm PSL}_n(q)$ is bounded by $l$. We generalize the Hull--Osin dichotomy for highly transitive permutation groups to linear groups over finite fields. Finally, we show that the rank limit of ${\rm GL}_n(q)$ for $q$ fixed and $n \to \infty$ is mixed identity free.
Source: arXiv (Cornell University)
Authors: Bradford, Henry, Schneider, Jakob, Thom, Andreas
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4388514944