Systoles of Arithmetic Hyperbolic 2- and 3-Manifolds

Rainie Bozzai; Benjamin Linowitz

doi:10.5802/afst.1793

Rainie Bozzai ¹ ; Benjamin Linowitz ²

¹ Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195
² Department of Mathematics, 10 North Professor Street, Oberlin, OH 44074

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 4, pp. 997-1017.

Résumé
Abstract

Dans cet article, nous étudions les systoles des variétés hyperboliques arithmétiques en dimension $2$ et $3$ . Notre premier résultat est la construction d’une infinité de variétés hyperboliques arithmétiques qui sont incommensurables deux-à-deux, ont toutes la même systole, et dont les volumes sont explicitement bornés. Notre deuxième résultat suppose fixé $x > 0$ et donne une borne supérieure pour le plus petit volume d’une variété hyperbolique arithmétique avec une systole supérieure à $x$ . Nous concluons en déterminant, pour certaines petites valeurs de $x$ , le plus petit volume d’une variété hyperbolique arithmétique principale sur $ℚ$ (en dimension $2$ ) ou sur $ℚ (i)$ (en dimension $3$ ) avec une systole supérieure à $x$ .

In this paper we study the systoles of arithmetic hyperbolic $2$ - and $3$ -manifolds. Our first result is the construction of infinitely many arithmetic hyperbolic $2$ - and $3$ -manifolds which are pairwise noncommensurable, all have the same systole, and whose volumes are explicitly bounded. Our second result fixes a positive number $x$ and gives an upper bound for the least volume of an arithmetic hyperbolic $2$ - or $3$ -manifold whose systole is greater than $x$ . We conclude by determining, for certain small values of $x$ , the least volume of a principal arithmetic hyperbolic $2$ -manifold over $ℚ$ or $3$ -manifold over $ℚ (i)$ whose systole is greater than $x$ .

Reçu le : 2022-04-21
Accepté le : 2023-06-09
Publié le : 2025-02-03

DOI : 10.5802/afst.1793

Affiliations des auteurs :

Rainie Bozzai ¹ ; Benjamin Linowitz ²

¹ Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195
² Department of Mathematics, 10 North Professor Street, Oberlin, OH 44074

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Rainie Bozzai and Benjamin Linowitz},
     title = {Systoles of {Arithmetic} {Hyperbolic} 2- and {3-Manifolds}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {997--1017},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Rainie Bozzai; Benjamin Linowitz. Systoles of Arithmetic Hyperbolic 2- and 3-Manifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 4, pp. 997-1017. doi : 10.5802/afst.1793. https://afst.centre-mersenne.org/articles/10.5802/afst.1793/

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