Number Theory/Geometry
Congruence obstructions to pseudomodularity of Fricke groups
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 603-606.

A pseudomodular group is a finite coarea non-arithmetic Fuchsian group whose set of cusps is P1(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 P1(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.

Published online:
DOI: 10.1016/j.crma.2008.04.005
Fithian, David 1

1 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
     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},
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Fithian, David. 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.

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