On the convergence of arithmetic orbifolds

Raimbault, Jean

Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2547-2596.

Cet article est consacré à l’étude de la géométrie globale de certaines orbi-variétés localement isométriques à un produit d’espaces tridimensionnels et de plans hyperboliques. On démontre que pour les petites dimensions (pour l’espace ou le plan hyperbolique, ou un produit de plans hyperboliques) certaines suites de telles orbi-variétés non-compactes de volume fini convergent vers l’espace symétrique en un sens géométrique précis (« convergence de Benjamini–Schramm »). On traite aussi le cas des réseaux arithmétiques maximaux en dimension trois dont les corps de traces sont quadratiques ou cubiques. Une des principales motivations est d’étudier l’asymptotique des nombres de Betti des groupes de Bianchi.

We discuss the geometry of some arithmetic orbifolds locally isometric to a product $X$ of real hyperbolic spaces ${ℍ}^m$ of dimension $m=2,3$, and prove that certain sequences of non-compact orbifolds are convergent to $X$ in a geometric (“Benjamini–Schramm”) sense for low-dimensional cases (when $X$ is equal to ${ℍ}^2\times {ℍ}^2$ or ${ℍ}^3$). We also deal with sequences of maximal arithmetic three–dimensional hyperbolic lattices defined over a quadratic or cubic field. A motivating application is the study of Betti numbers of Bianchi groups.

Classification : 22E40, 11F75, 11F72, 57M27
Keywords: Arithmetic hyperbolic manifolds, Limit multiplicities, Three–dimensional manifolds
Mot clés : Variétés hyperboliques arithmétiques, Multiplicités limites, Variétés tridimensionnelles

[1] Abért, Miklós; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolov, Nikolay; Raimbault, Jean; Samet, Iddo On the growth of $L^2$-invariants for sequences of lattices in Lie groups, Ann. Math., Volume 185 (2017) no. 3, pp. 711-790 | DOI | Zbl

[2] Abért, Miklós; Bergeron, Nicolas; Virág, Bálint Convergence of weakly Ramanujan locally symmetric spaces (in preparation)

[3] Abért, Miklós; Glasner, Yair; Virág, Bálint Kesten’s theorem for invariant random subgroups, Duke Math. J., Volume 163 (2014) no. 3, pp. 465-488 | DOI | MR | Zbl

[4] Agol, Ian The virtual Haken conjecture, Doc. Math., J. DMV, Volume 18 (2013), pp. 1045-1087 (With an appendix by Agol, Daniel Groves, and Jason Manning) | MR | Zbl

[5] Bachman, David; Cooper, Daryl; White, Matthew E. Large embedded balls and Heegaard genus in negative curvature, Algebr. Geom. Topol., Volume 4 (2004), pp. 31-47 | DOI | MR | Zbl

[6] Benjamini, Itai; Schramm, Oded Recurrence of distributional limits of finite planar graphs, Electron. J. Probab., Volume 6 (2001) paper no. 23, 13 pp. (electronic) | DOI | MR | Zbl

[7] Bergeron, Nicolas Le spectre des surfaces hyperboliques, Savoirs Actuels (Les Ulis), EDP Sciences, Les Ulis, 2011, x+338 pages | MR | Zbl

[8] Bhargava, Manjul The density of discriminants of quartic rings and fields, Ann. Math., Volume 162 (2005) no. 2, pp. 1031-1063 | DOI | MR | Zbl

[9] Bhargava, Manjul; Shankar, Arul; Taniguchi, Takashi; Thorne, Frank; Tsimerman, Jacob; Zhao, Yongqiang Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves (2017) (https://arxiv.org/abs/1701.02458)

[10] Bianchi, Luigi Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginari, Math. Ann., Volume 40 (1892) no. 3, pp. 332-412 | DOI | MR | Zbl

[11] Biringer, Ian; Souto, Juan A finiteness theorem for hyperbolic 3-manifolds, J. Lond. Math. Soc., Volume 84 (2011) no. 1, pp. 227-242 | DOI | MR | Zbl

[12] Bolte, Jens; Johansson, Stefan A spectral correspondence for Maaß waveforms, Geom. Funct. Anal., Volume 9 (1999) no. 6, pp. 1128-1155 | DOI | MR | Zbl

[13] Borel, Armand Commensurability classes and volumes of hyperbolic $3$-manifolds, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 8 (1981) no. 1, pp. 1-33 | MR | Zbl

[14] Borel, Armand; Prasad, Gopal Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math., Inst. Hautes Étud. Sci. (1989) no. 69, pp. 119-171 | DOI | MR | Zbl

[15] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999, xxii+643 pages | DOI | MR | Zbl

[16] Bump, Daniel Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, Cambridge, 1997, xiv+574 pages | DOI | MR | Zbl

[17] Clozel, Laurent Démonstration de la conjecture $\tau$, Invent. Math., Volume 151 (2003) no. 2, pp. 297-328 | DOI | MR | Zbl

[18] Deraux, Martin; Parker, John R.; Paupert, Julien New non-arithmetic complex hyperbolic lattices, Invent. Math., Volume 203 (2016) no. 3, pp. 681-771 | DOI | MR | Zbl

[19] Efrat, Isaac Y. The Selberg trace formula for ${\mathrm{PSL}}_2{\left(ℝ\right)}^n$, Mem. Am. Math. Soc., Volume 359 (1987) no. 359, pp. 1-111 | DOI | MR | Zbl

[20] Einsiedler, Manfred; Lindenstrauss, Elon; Michel, Philippe; Venkatesh, Akshay Distribution of periodic torus orbits and Duke’s theorem for cubic fields, Ann. Math., Volume 173 (2011) no. 2, pp. 815-885 | DOI | MR | Zbl

[21] Elkies, Noam D. Fundamental units of imaginary quartic fields, 2013 (http://mathoverflow.net/q/137480, accepted answer to a question by Jean Raimbault)

[22] Ellenberg, Jordan S.; Venkatesh, Akshay Reflection principles and bounds for class group torsion, Int. Math. Res. Not. IMRN (2007) no. 1 (Art. ID rnm002, 18 pp.) | DOI | MR | Zbl

[23] Fröhlich, A.; Taylor, Martin J. Algebraic number theory, Cambridge Studies in Advanced Mathematics, 27, Cambridge University Press, 1993, xiv+355 pages | MR | Zbl

[24] Gendulphe, Matthieu Systole et rayon interne des variétés hyperboliques non compactes, Geom. Topol., Volume 19 (2015) no. 4, pp. 2039-2080 | DOI | MR | Zbl

[25] Godement, Roger Domaines fondamentaux des groupes arithmétiques (Séminaire Bourbaki), Volume 8, Société Mathématique de France, 1964, pp. 201-225 | MR | Zbl

[26] Gromov, Mikhael Leonidovich; Guth, Larry Generalizations of the Kolmogorov-Barzdin embedding estimates, Duke Math. J., Volume 161 (2012) no. 13, pp. 2549-2603 | DOI | MR | Zbl

[27] Gromov, Mikhael Leonidovich; Piatetski-Shapiro, Ilya I. Non-arithmetic groups in Lobachevsky spaces, Publ. Math., Inst. Hautes Étud. Sci. (1988) no. 66, pp. 93-103 | MR | Zbl

[28] Lenstra, Hendrik W.jun. Algorithms in algebraic number theory, Bull. Am. Math. Soc., Volume 26 (1992) no. 2, pp. 211-244 | DOI | MR | Zbl

[29] Lubotzky, Alexander; Segal, Dan Subgroup growth, Progress in Mathematics, 212, Birkhäuser, Basel, 2003, xxii+453 pages | DOI | MR | Zbl

[30] Maclachlan, Colin; Reid, Alan W. The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, 219, Springer, New York, 2003, xiv+463 pages | MR | Zbl

[31] Müller, Werner; Pfaff, Jonathan The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume, J. Funct. Anal., Volume 267 (2014) no. 8, pp. 2731-2786 | DOI | MR | Zbl

[32] Ohno, Shin; Watanabe, Takao Estimates of Hermite constants for algebraic number fields, Comment. Math. Univ. St. Pauli, Volume 50 (2001) no. 1, pp. 53-63 | MR | Zbl

[33] Otal, Jean-Pierre Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque, 235, Société Mathématique de France, 1996, x+159 pages | MR | Zbl

[34] Petersen, Peter Riemannian geometry, Graduate Texts in Mathematics, 171, Springer, 2006, xvi+401 pages | MR | Zbl

[35] Raghunathan, Madabusi Santanam Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 68, Springer, 1972, ix+227 pages | MR | Zbl

[36] Rahm, Alexander D. Higher torsion in the Abelianization of the full Bianchi groups, LMS J. Comput. Math., Volume 16 (2013), pp. 344-365 | DOI | MR | Zbl

[37] Rahm, Alexander D.; Şengün, Mehmet Haluk On level one cuspidal Bianchi modular forms, LMS J. Comput. Math., Volume 16 (2013), pp. 187-199 | DOI | MR | Zbl

[38] Raimbault, Jean Analytic, Reidemeister and homological torsion for congruence three–manifolds (2013) (https://arxiv.org/abs/1307.2845)

[39] Raimbault, Jean Asymptotics of analytic torsion for hyperbolic three–manifolds (2013) (https://arxiv.org/abs/1212.3161)

[40] Raimbault, Jean A note on maximal lattice growth in $\mathrm{SO}(1,n)$, Int. Math. Res. Not. IMRN, Volume 2013 (2013) no. 16, pp. 3722-3731 | DOI | MR | Zbl

[41] Rohlfs, Jürgen Die maximalen arithmetisch definierten Untergruppen zerfallender einfacher Gruppen, Math. Ann., Volume 244 (1979) no. 3, pp. 219-231 | DOI | MR | Zbl

[42] Rohlfs, Jürgen On the cuspidal cohomology of the Bianchi modular groups, Math. Z., Volume 188 (1985) no. 2, pp. 253-269 | DOI | MR | Zbl

[43] Sarnak, Peter C. The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math., Volume 151 (1983) no. 3-4, pp. 253-295 | DOI | MR | Zbl

[44] Serre, Jean-Pierre Arbres, amalgames, ${\mathrm{SL}}_2$, Astérisque, 46, Société Mathématique de France, 1977, 189 pages | MR | Zbl

[45] Siegel, Carl Ludwig Lectures on the geometry of numbers, Springer, 1989, x+160 pages | DOI | MR | Zbl

[46] Vignéras, Marie-France Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, 800, Springer, Berlin, 1980, vii+169 pages | MR | Zbl