Comptes Rendus
Research article - Algebra, Dynamical systems
Irrationality of the length spectrum
George Peterzil1; Guy Sapire1
1 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 825-828

It is a classical result of Dal’Bo that the length spectrum of a non-elementary Fuchsian group is non-arithmetic, namely, it generates a dense additive subgroup of $\mathbb{R}$. In this note we provide an elementary proof of an extension of this theorem: a non-elementary Fuchsian group contains two elements whose lengths are linearly independent over $\mathbb{Q}$, reproving a result of Prasad and Rapinchuk [9].

C’est un résultat classique de Dal’Bo que le spectre des longueurs d’un groupe fuchsien non élémentaire est non arithmétique, c’est-à-dire qu’il génère un sous-groupe additif dense de $\mathbb{R}$. Dans cette note, nous fournissons une preuve élémentaire d’une extension de ce théorème : un groupe fuchsien non élémentaire contient deux éléments dont les longueurs sont linéairement indépendantes sur $\mathbb{Q}$, reproduisant un résultat de Prasad et Rapinchuk [9] dans un cas particulier.

Received:
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Accepted:
Published online:
DOI: 10.5802/crmath.756
Classification: 20H10, 37D40
Keywords: Length spectrum, Fuchsian group, trace field
Mots-clés : Spectre de longueur, groupe fuchsien, corps de traces

George Peterzil 1; Guy Sapire 1

1 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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George Peterzil; Guy Sapire. Irrationality of the length spectrum. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 825-828. doi: 10.5802/crmath.756

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