Congruence obstructions to pseudomodularity of Fricke groups

David Fithian

doi:10.1016/j.crma.2008.04.005

Number Theory/Geometry

Congruence obstructions to pseudomodularity of Fricke groups

Presented by: Jean-Pierre Serre

David Fithian ¹

¹ Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA

Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 603-606.

Abstract
Résumé

A pseudomodular group is a finite coarea non-arithmetic Fuchsian group whose set of cusps is $P^{1} (Q)$ . 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 $Q$ , 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 $P^{1} (Q)$ . 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 $Q$ . Nous en déduisons, l'existence d'une infinité de groupes de Fricke à pointes rationnelles, qui ne sont ni pseudo-modulaires ni arithmétiques.

Received: 2007-08-16
Accepted: 2008-04-14
Published online: 2008-05-15

DOI: 10.1016/j.crma.2008.04.005

Author's affiliations:

David Fithian ¹

¹ Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA

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     title = {Congruence obstructions to pseudomodularity of {Fricke} groups},
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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/

References
Cited by

[1] R. Abe 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] A.F. Beardon The Geometry of Discrete Groups, Graduate Texts in Math., vol. 91, Springer-Verlag, 1983

[3] C. Culler; P. Shalen Varieties of group representations and splittings of 3-manifolds, Ann. of Math., Volume 117 (1983) no. 1, pp. 109-146

[4] D.D. Long; A.W. Reid Generalized Dedekind sums, Proceedings of the Casson Fest, Geometry and Topology Monographs, vol. 7, 2004, pp. 205-212

[5] D.D. Long; A.W. Reid Pseudomodular surfaces, J. Reine Angew. Math., Volume 552 (2002), pp. 77-100

[6] G. Shimura Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1971

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