[Obstacles à la pseudo-modularité des groupes de Fricke données par des conditions de congruence]
Un groupe pseudo-modulaire est un groupe fuchsien, non-arithmétique et de coaire finie dont l'ensemble des pointes est . Long et Reid en ont construit un nombre fini en considérant les groupes fuchsiens qui uniformisent les tores à un trou, appelés groupes de Fricke. Nous démontrons ici qu'un groupe de Fricke, dont les pointes sont les nombres rationnels et l'infini, est pseudo-modulaire si et seulement si l'ensemble de ses pointes finies est dense dans le groupe des adèles finies de . Nous en déduisons, l'existence d'une infinité de groupes de Fricke à pointes rationnelles, qui ne sont ni pseudo-modulaires ni arithmétiques.
A pseudomodular group is a finite coarea non-arithmetic Fuchsian group whose set of cusps is . Long and Reid constructed finitely many of these by considering Fuchsian groups uniformizing one-cusped tori, i.e., Fricke groups. We show that a zonal (i.e., having a cusp at infinity) Fricke group with rational cusps is pseudomodular if and only if its set of finite cusps is dense in the finite adeles of , and that there are infinitely many Fricke groups with rational cusps that are neither pseudomodular nor arithmetic.
Accepté le :
Publié le :
@article{CRMATH_2008__346_11-12_603_0,
author = {Fithian, David},
title = {Congruence obstructions to pseudomodularity of {Fricke} groups},
journal = {Comptes Rendus. Math\'ematique},
pages = {603--606},
publisher = {Elsevier},
volume = {346},
number = {11-12},
year = {2008},
doi = {10.1016/j.crma.2008.04.005},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.04.005/}
}
TY - JOUR AU - Fithian, David TI - Congruence obstructions to pseudomodularity of Fricke groups JO - Comptes Rendus. Mathématique PY - 2008 SP - 603 EP - 606 VL - 346 IS - 11-12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.04.005/ DO - 10.1016/j.crma.2008.04.005 LA - en ID - CRMATH_2008__346_11-12_603_0 ER -
%0 Journal Article %A Fithian, David %T Congruence obstructions to pseudomodularity of Fricke groups %J Comptes Rendus. Mathématique %D 2008 %P 603-606 %V 346 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.04.005/ %R 10.1016/j.crma.2008.04.005 %G en %F CRMATH_2008__346_11-12_603_0
Fithian, David. Congruence obstructions to pseudomodularity of Fricke groups. Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 603-606. doi : 10.1016/j.crma.2008.04.005. http://www.numdam.org/articles/10.1016/j.crma.2008.04.005/
[1] On correspondences between once punctured tori and closed tori: Fricke groups and real lattices, Tokyo J. Math., Volume 23 (2000) no. 2, pp. 269-293
[2] The Geometry of Discrete Groups, Graduate Texts in Math., vol. 91, Springer-Verlag, 1983
[3] Varieties of group representations and splittings of 3-manifolds, Ann. of Math., Volume 117 (1983) no. 1, pp. 109-146
[4] Generalized Dedekind sums, Proceedings of the Casson Fest, Geometry and Topology Monographs, vol. 7, 2004, pp. 205-212
[5] Pseudomodular surfaces, J. Reine Angew. Math., Volume 552 (2002), pp. 77-100
[6] Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1971
Cité par Sources :
