Density of systoles of hyperbolic manifolds

Sami Douba; Junzhi Huang

doi:10.5802/crmath.689

Article de recherche - Géométrie et Topologie, Théorie des groupes

Density of systoles of hyperbolic manifolds
[Densité de systoles de variétés hyperboliques]

Sami Douba ¹ ; Junzhi Huang ²

¹ Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France
² Department of Mathematics, Yale University, New Haven, CT 06511, USA

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1819-1824.

Abstract (VO)
Résumé

We show that for each $n \geq 2$ , the systoles of closed hyperbolic $n$ -manifolds form a dense subset of $(0, + \infty)$ . We also show that for any $n \geq 2$ and any Salem number $λ$ , there is a closed arithmetic hyperbolic $n$ -manifold of systole $log (λ)$ . In particular, the Salem conjecture holds if and only if the systoles of closed arithmetic hyperbolic manifolds in some (any) dimension fail to be dense in $(0, + \infty)$ .

Nous démontrons que, pour tout $n \geq 2$ , les systoles de variétés hyperboliques compactes sans bord de dimension $n$ constituent une partie dense de $] 0, + \infty [$ . Nous démontrons de plus que, pour tout $n \geq 2$ et tout nombre de Salem $λ$ , il existe une variété hyperbolique arithmétique compacte sans bord de dimension $n$ et de systole $log (λ)$ . En particulier, la conjecture de Salem est vraie si et seulement si les systoles de variétés hyperboliques arithmétiques compactes sans bord d’une certaine dimension (de manière équivalente, de dimension quelconque) ne sont pas denses dans $] 0, + \infty [$ .

Reçu le : 2024-06-05
Accepté le : 2024-09-17
Publié le : 2024-11-25

Zbl

DOI : 10.5802/crmath.689

Classification : 22E40, 11F06
Keywords: Geometric topology, hyperbolic manifolds, systoles, arithmetic groups
Mots-clés : Topologie géométrique, variétés hyperboliques, systoles, groupes arithmétiques

Affiliations des auteurs :

Sami Douba ¹ ; Junzhi Huang ²

¹ Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France
² Department of Mathematics, Yale University, New Haven, CT 06511, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Sami Douba and Junzhi Huang},
     title = {Density of systoles of hyperbolic manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1819--1824},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.689},
     zbl = {07949989},
     language = {en},
}

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Sami Douba; Junzhi Huang. Density of systoles of hyperbolic manifolds. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1819-1824. doi : 10.5802/crmath.689. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.689/

Bibliographie
Cité par

[1] Ian Agol Systoles of hyperbolic 4-manifolds (2006) (https://arxiv.org/abs/math/0612290)

[2] Ian Agol; Darren D. Long; Alan W. Reid The Bianchi groups are separable on geometrically finite subgroups, Ann. Math., Volume 153 (2001) no. 3, pp. 599-621 | DOI | MR | Zbl

[3] Emmanuel Breuillard; Bertrand Deroin Salem numbers and the spectrum of hyperbolic surfaces, Int. Math. Res. Not., Volume 2020 (2020) no. 22, pp. 8234-8250 | DOI | MR | Zbl

[4] Nicolas Bergeron Premier nombre de Betti et spectre du laplacien de certaines variétés hyperboliques, Enseign. Math., Volume 46 (2000) no. 1-2, pp. 109-137 | MR | Zbl

[5] Armand Borel; Harish-Chandra Arithmetic subgroups of algebraic groups, Ann. Math., Volume 75 (1962), pp. 485-535 | DOI | MR | Zbl

[6] Nicolas Bergeron; Frédéric Haglund; Daniel T. Wise Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc., Volume 83 (2011) no. 2, pp. 431-448 | DOI | MR | Zbl

[7] Riccardo Benedetti; Carlo Petronio Lectures on hyperbolic geometry, Universitext, Springer, 1992, xiv+330 pages | DOI | MR | Zbl

[8] Mikhail V. Belolipetsky; Scott A. Thomson Systoles of hyperbolic manifolds, Algebr. Geom. Topol., Volume 11 (2011) no. 3, pp. 1455-1469 | DOI | MR | Zbl

[9] Gregory Cosac; Cayo Dória Closed geodesics on semi-arithmetic Riemann surfaces, Math. Res. Lett., Volume 29 (2022) no. 4, pp. 961-1001 | DOI | MR | Zbl

[10] Sami Douba Systoles of hyperbolic hybrids (2023) (https://arxiv.org/abs/2309.16051)

[11] Vincent Emery; John G. Ratcliffe; Steven T. Tschantz Salem numbers and arithmetic hyperbolic groups, Trans. Am. Math. Soc., Volume 372 (2019) no. 1, pp. 329-355 | DOI | MR | Zbl

[12] Mikolaj Fraczyk; Lam L. Pham Bottom of the length spectrum of arithmetic orbifolds, Trans. Am. Math. Soc., Volume 376 (2023) no. 7, pp. 4745-4764 | DOI | MR | Zbl

[13] Tsachik Gelander Homotopy type and volume of locally symmetric manifolds, Duke Math. J., Volume 124 (2004) no. 3, pp. 459-515 | DOI | MR | Zbl

[14] Petr Kravchuk; Dalimil Mazáč; Sridip Pal Automorphic spectra and the conformal bootstrap, Commun. Am. Math. Soc., Volume 4 (2024), pp. 1-63 | DOI | MR | Zbl

[15] Derrick H. Lehmer Factorization of certain cyclotomic functions, Ann. Math., Volume 34 (1933) no. 3, pp. 461-479 | DOI | MR | Zbl

[16] Darren D. Long Immersions and embeddings of totally geodesic surfaces, Bull. Lond. Math. Soc., Volume 19 (1987) no. 5, pp. 481-484 | DOI | MR | Zbl

[17] Michael Magee The limit points of the bass notes of arithmetic hyperbolic surfaces (2024) (https://arxiv.org/abs/2403.00928)

[18] Gregory A. Margulis Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 17, Springer, 1991, x+388 pages | DOI | MR | Zbl

[19] George D. Mostow Quasi-conformal mappings in $n$ -space and the rigidity of hyperbolic space forms, Publ. Math., Inst. Hautes Étud. Sci., Volume 34 (1968), pp. 53-104 | DOI | Numdam | MR | Zbl

[20] Walter D. Neumann; Alan W. Reid Arithmetic of hyperbolic manifolds, Topology ’90 (Columbus, OH, 1990) (Ohio State University Mathematical Research Institute Publications), Volume 1, Walter de Gruyter, 1992, pp. 273-310 | MR | Zbl

[21] Raphaël Salem Algebraic numbers and Fourier analysis, Selected reprints (The Wadsworth Mathematics Series), Wadsworth, 1983, p. iii+68 | MR | Zbl

[22] Peter Scott Subgroups of surface groups are almost geometric, J. Lond. Math. Soc., Volume 17 (1978) no. 3, pp. 555-565 | DOI | MR | Zbl

[23] Scott A. Thomson Quasi-arithmeticity of lattices in $PO (n, 1)$ , Geom. Dedicata, Volume 180 (2016), pp. 85-94 | DOI | MR | Zbl

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