Non-singular word maps for linear groups
| Autor: | Bradford, Henry, Schneider, Jakob, Thom, Andreas |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 17 pages, no figures |
| Databáze: | arXiv |
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