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.
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.
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David Fithian 1
@article{CRMATH_2008__346_11-12_603_0, author = {David Fithian}, 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}, }
David Fithian. Congruence obstructions to pseudomodularity of Fricke groups. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 603-606. doi : 10.1016/j.crma.2008.04.005. https://comptes-rendus.academie-sciences.fr/mathematique/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
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