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.
Mots-clés : Fuchsian groups, commensurability
Greg McShane 1

@article{AFST_2019_6_28_3_471_0, author = {Greg McShane}, title = {Geodesic intersections and isoxial {Fuchsian} groups.}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {471--489}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 28}, number = {3}, year = {2019}, doi = {10.5802/afst.1606}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1606/} }
TY - JOUR AU - Greg McShane TI - Geodesic intersections and isoxial Fuchsian groups. JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2019 SP - 471 EP - 489 VL - 28 IS - 3 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1606/ DO - 10.5802/afst.1606 LA - en ID - AFST_2019_6_28_3_471_0 ER -
%0 Journal Article %A Greg McShane %T Geodesic intersections and isoxial Fuchsian groups. %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2019 %P 471-489 %V 28 %N 3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1606/ %R 10.5802/afst.1606 %G en %F AFST_2019_6_28_3_471_0
Greg McShane. Geodesic intersections and isoxial Fuchsian groups.. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume spécial en l’honneur de Jean-Pierre OTAL “Low dimensional topology, hyperbolic manifolds and spectral geometry”, 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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