On the surjectivity of Engel words on \(PSL(2,q)\)
type: Item
instance of: scholarly article
title: On the surjectivity of Engel words on \(PSL(2,q)\)
zbMATH Open document ID: 1261.14010
DOI: 10.4171/GGD/162
author: Tatiana Bandman
author: Shelly Garion
author: Fritz Grunewald
published in: Groups, Geometry, and Dynamics
publication date: 2012-10-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.1397
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 37P25
Mathematics Subject Classification ID: 37P55
zbMATH DE Number: 6088867
zbMATH Keywords: Engel words
zbMATH Keywords: special linear group
zbMATH Keywords: arithmetic dynamics
zbMATH Keywords: periodic points
zbMATH Keywords: finite fields
zbMATH Keywords: trace map
describes a project that uses: Magma
MaRDI profile type: Publication
arXiv ID: 1008.1397
cites work: Q4336964
Recommended article: On surjectivity of word maps on \(\mathrm{PSL}_2\)
Recommended article: SURJECTIVITY AND EQUIDISTRIBUTION OF THE WORD xayb ON PSL(2, q) AND SL(2, q)
Recommended article: Surjectivity of certain word maps on \(\mathrm{PSL}(2,\mathbb {C})\) and \(\mathrm{SL}(2,\mathbb {C})\)
Recommended article: On the surjectivity of certain word maps on SU(2)
Recommended article: Surjective word maps and Burnside's \(p^aq^b\) theorem
Recommended article: Recursive sequences of surjective word maps for the algebraic groups \(\mathrm{PGL}_2\) and \({{\mathrm{SL}}}_2\)
Recommended article: On a problem of P. Hall for Engel words. II.
Recommended article: On a problem of P. Hall for Engel words.
Recommended article: A note on Engel words of finite groups
Recommended article: On surjective quadratic maps of \(\mathbb{P}^2 \)
review text: The paper presents several results regarding the \(n\)-th Engel word map on \(\text{PSL}(2,q)\) and \(\text{SL}(2,q)\), where \(n\) and \(q\) range through the natural numbers. The Engel word maps are recursively defined generalizations of the commutator word map, and the results in the paper are extensions of previously know results about the commutator word map. The authors note that a key difficulty in moving beyond the commutator word map is that there is no (known) quantity for Engel words that is analogous to the character sum for the commutator word. The main result is that, like the commutator word map, the \(n\)-th Engel word map on \(\text{PSL}(2,q)\) is surjective in many cases and should be expected to always be surjective. It is also shown, however, that the \(n\)-th Engel word map on \(\text{SL}(2,q)\) is not even almost surjective in infinitely many cases (in contrast to the situation for the commutator map). The paper also includes results about almost equidistribution and almost measure-preservation for the Engel word maps on \(\text{PSL}(2,q)\).NEWLINENEWLINEThe main tool used in the paper is a so-called trace map on an affine space whose definition is such that surjectivity of the \(n\)-th Engel word map on \(\text{PSL}(2,q)\) corresponds to surjectivity of the \(n\)-th iterate of the trace map. Thus the results are achieved via the dynamics of the trace map. The proof of the main result relies on a nice accounting of points on certain affine curves defined over finite fields.
description: scientific article; zbMATH DE number 6088867