Geodesic intersections and isoxial Fuchsian groups.
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 3, pp. 471-489.

The set of axes of hyperbolic elements in a Fuchsian group depends on the commensurability class of the group. In fact, it has been conjectured that it determines the commensurability class and this has been verified for groups of the second kind by G. Mess and for arithmetic groups by D. Long and A. Reid. Here we show that the conjecture holds for almost all Fuchsian groups and explain why our method fails for arithmetic groups.

L’ensemble des axes d’éléments hyperboliques dans un groupe fuchsien dépend de la classe de commensurabilité du groupe. En effet, cet ensemble détermine la classe de commensurabilité pour les groupes du deuxième type, d’après G. Mess, et pour les groupes arithmétiques, d’après D. Long et A. Reid. Selon une veille conjecture, la classe de commensurabilité d’un groupe fuchsien non élémentaire est toujours déterminée par ses axes. Nous montrons ici que la conjecture est vraie pour presque tous les groupes fuchsiens et expliquons pourquoi notre méthode ne s’applique pas aux groupes arithmétiques.

Published online:
DOI: 10.5802/afst.1606
Classification: 57M27, 37E30, 57M55
Keywords: Fuchsian groups, commensurability

Greg McShane 1

1 UFR de Mathématiques Institut Fourier 100 rue des maths BP 74, 38402 St-Martin-d’Hères cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Greg McShane. Geodesic intersections and isoxial Fuchsian groups.. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 3, pp. 471-489. doi : 10.5802/afst.1606. https://afst.centre-mersenne.org/articles/10.5802/afst.1606/

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