Geodesic
A height gap in $\mathrm{GL}_{d}(\overline{\mathbb{Q}})$ and almost laws
Lvzhou Chen1 orcid ; Sebastian Hurtado2 ; Homin Lee3 orcid
1Purdue University, West Lafayette, USA
2Yale University, New Haven, USA
3Northwestern University, Evanston, USA
Groups, geometry, and dynamics, Tome 19 (2025) no. 3, pp. 899-912

Voir la notice de l'article provenant de la source EMS Press

DOI

E. Breuillard showed that finite subsets F of matrices in GLd​(Q​) generating non-virtually solvable groups have normalized height h^(F)≥εd​, for some positive εd​>0. The normalized height h^(F) is a measure of the arithmetic size of F and this result can be thought of as a non-abelian analog of Lehmer’s Mahler measure problem. We give a new shorter proof of this result. Our key idea relies on the existence of particular word maps in compact Lie groups (known as almost laws) whose image lies close to the identity element.
DOI : 10.4171/ggd/800
Classification : 20G25, 20G30, 20E07
Mots-clés : height gap, almost law, word maps, arithmetic groups, and linear groups

Lvzhou Chen  1   ; Sebastian Hurtado  2   ; Homin Lee  3

1 Purdue University, West Lafayette, USA
2 Yale University, New Haven, USA
3 Northwestern University, Evanston, USA 
Lvzhou Chen; Sebastian Hurtado; Homin Lee. A height gap in $\mathrm{GL}_{d}(\overline{\mathbb{Q}})$ and almost laws. Groups, geometry, and dynamics, Tome 19 (2025) no. 3, pp. 899-912. doi: 10.4171/ggd/800
@article{10_4171_ggd_800,
     author = {Lvzhou Chen and Sebastian Hurtado and Homin Lee},
     title = {A height gap in $\mathrm{GL}_{d}(\overline{\mathbb{Q}})$ and almost laws},
     journal = {Groups, geometry, and dynamics},
     pages = {899--912},
     year = {2025},
     volume = {19},
     number = {3},
     doi = {10.4171/ggd/800},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/800/}
}
TY  - JOUR
AU  - Lvzhou Chen
AU  - Sebastian Hurtado
AU  - Homin Lee
TI  - A height gap in $\mathrm{GL}_{d}(\overline{\mathbb{Q}})$ and almost laws
JO  - Groups, geometry, and dynamics
PY  - 2025
SP  - 899
EP  - 912
VL  - 19
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4171/ggd/800/
DO  - 10.4171/ggd/800
ID  - 10_4171_ggd_800
ER  - 
%0 Journal Article
%A Lvzhou Chen
%A Sebastian Hurtado
%A Homin Lee
%T A height gap in $\mathrm{GL}_{d}(\overline{\mathbb{Q}})$ and almost laws
%J Groups, geometry, and dynamics
%D 2025
%P 899-912
%V 19
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/800/
%R 10.4171/ggd/800
%F 10_4171_ggd_800

Cité par Sources :