Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

Linowitz, Benjamin; McReynolds, D.B.; Pollack, Paul; Thompson, Lola

Comptes Rendus. Mathématique, Tome 355 (2017) no. 11, pp. 1121-1126.

En 1992, Reid a demandé si deux 3-variétés hyperboliques partageant le même spectre de longueurs géodésiques sont nécessairement commensurables. Ceci s'avère être vrai quand les variétés sont arithmétiques, mais la question reste ouverte dans le cas non arithmétique. Comme premier pas vers une réponse négative à cette question, Futer et Millichap ont récemment construit un nombre infini de paires de 3-variétés hyperboliques non arithmétiques et non commensurables ayant le même volume et dont les spectres de longueurs commencent avec les mêmes m longueurs géodésiques. Dans le présent article, nous démontrons que ce phénomène est étonnamment commun dans le contexte arithmétique. En particulier, étant donné une 3-variété hyperbolique arithmétique dérivée d'une algèbre de quaternions, un sous-ensemble fini S de son spectre de longueurs géodésiques et un entier k≥2, nous construisons un nombre infini de k-tuples de 3-variétés hyperboliques arithmétiques qui sont non commensurables deux à deux, ont un spectre de longueurs géodésiques contenant S et dont le volume appartient à un intervalle de longueur bornée (cette borne est, en outre, universelle pour chaque entier k). Notre preuve s'appuie sur un résultat sur les petits écarts entre idéaux premiers d'un corps de nombres appartenant à un ensemble de Chebotarev ; ce résultat généralise un article récent de Thorner.

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first m geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset S of its geodesic length spectrum, and any k≥2, we produce infinitely many k-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing S, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.

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