We investigate the surjectivity of the word map defined by the n-th Engel word on the groups PSL(2,q) and SL(2,q). For SL(2,q) we show that this map is surjective onto the subset SL(2,q)∖{−id}⊂SL(2,q) provided that q≥q0(n) is sufficiently large. Moreover, we give an estimate for q0(n). We also present examples demonstrating that this does not hold for all q. We conclude that the n-th Engel word map is surjective for the groups PSL(2,q) when q≥q0(n). By using a computer, we sharpen this result and show that for any n≤4 the corresponding map is surjective for all the groups PSL(2,q). This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the n-th Engel word map is almost measure-preserving for the family of groups PSL(2,q), with q odd, answering another question of Shalev.
1
Bar-Ilan University, Ramat Gan, Israel
2
Universität Münster, Germany
3
Heinrich-Heine-Universität, Düsseldorf, Germany
Tatiana Bandman; Shelly Garion; Fritz Grunewald. On the surjectivity of Engel words on PSL(2,$q$). Groups, geometry, and dynamics, Tome 6 (2012) no. 3, pp. 409-439. doi: 10.4171/ggd/162
@article{10_4171_ggd_162,
author = {Tatiana Bandman and Shelly Garion and Fritz Grunewald},
title = {On the surjectivity of {Engel} words on {PSL(2,}$q$)},
journal = {Groups, geometry, and dynamics},
pages = {409--439},
year = {2012},
volume = {6},
number = {3},
doi = {10.4171/ggd/162},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/162/}
}
TY - JOUR
AU - Tatiana Bandman
AU - Shelly Garion
AU - Fritz Grunewald
TI - On the surjectivity of Engel words on PSL(2,$q$)
JO - Groups, geometry, and dynamics
PY - 2012
SP - 409
EP - 439
VL - 6
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/162/
DO - 10.4171/ggd/162
ID - 10_4171_ggd_162
ER -
%0 Journal Article
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%T On the surjectivity of Engel words on PSL(2,$q$)
%J Groups, geometry, and dynamics
%D 2012
%P 409-439
%V 6
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/162/
%R 10.4171/ggd/162
%F 10_4171_ggd_162
