Analytic, Reidemeister and homological torsion for congruence three–manifolds

Jean Raimbault

doi:10.5802/afst.1605

Jean Raimbault ¹

¹ Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 3, pp. 417-469.

Résumé
Abstract

Soit $Γ$ un groupe de Bianchi. Pour certains $ℤ Γ$ -modules $V_{ℤ}$ , et suites $Γ_{n}$ de sous-groupes de congruence de $Γ$ nous démontrons une borne supérieure, conjecturée optimale, pour la taille du sous-groupe de torsion de l’homologie $H_{1} (Γ_{n}, V_{ℤ})$ . On démontre aussi des résultats de multiplicités limites pour les facteurs irréductibles des espaces $L_{cusp}^{2} (Γ_{n} ∖ {SL}_{2} (ℂ))$ .

For a given Bianchi group $Γ$ and certain natural coefficent modules $V_{ℤ}$ and sequences $Γ_{n}$ of congruence subgroups of $Γ$ we give a conjecturally optimal upper bound for the size of the torsion subgroup of $H_{1} (Γ_{n}; V_{ℤ})$ . We also prove limit multiplicity results for the irreducible components of $L_{cusp}^{2} (Γ_{n} ∖ {SL}_{2} (ℂ))$ .

Publié le : 2019-12-06

DOI : 10.5802/afst.1605

Classification : 11F75, 11F72, 22E40, 57M10
Mots clés : Congruence groups, hyperbolic manifolds, homology

Affiliations des auteurs :

Jean Raimbault ¹

¹ Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AFST_2019_6_28_3_417_0,
     author = {Jean Raimbault},
     title = {Analytic, {Reidemeister} and homological torsion for congruence three{\textendash}manifolds},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {417--469},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 28},
     number = {3},
     year = {2019},
     doi = {10.5802/afst.1605},
     mrnumber = {4014778},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1605/}
}

TY  - JOUR
AU  - Jean Raimbault
TI  - Analytic, Reidemeister and homological torsion for congruence three–manifolds
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2019
SP  - 417
EP  - 469
VL  - 28
IS  - 3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1605/
DO  - 10.5802/afst.1605
LA  - en
ID  - AFST_2019_6_28_3_417_0
ER  -

%0 Journal Article
%A Jean Raimbault
%T Analytic, Reidemeister and homological torsion for congruence three–manifolds
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2019
%P 417-469
%V 28
%N 3
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1605/
%R 10.5802/afst.1605
%G en
%F AFST_2019_6_28_3_417_0

Jean Raimbault. Analytic, Reidemeister and homological torsion for congruence three–manifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 3, pp. 417-469. doi : 10.5802/afst.1605. https://afst.centre-mersenne.org/articles/10.5802/afst.1605/

Bibliographie
Cité par

[1] Miklos Abert; Nicolas Bergeron; Ian Biringer; Tsachik Gelander; Nikolay Nikolov; Jean Raimbault; Iddo Samet 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 | MR | Zbl

[2] Tobias Berger Denominators of Eisenstein cohomology classes for ${GL}_{2}$ over imaginary quadratic fields, Manuscr. Math., Volume 125 (2008) no. 4, pp. 427-470 | DOI | MR | Zbl

[3] Nicolas Bergeron; Mehmet Haluk Şengün; Akshay Venkatesh Torsion homology growth and cycle complexity of arithmetic manifolds, Duke Math. J., Volume 165 (2016) no. 9, pp. 1629-1693 | DOI | MR | Zbl

[4] Nicolas Bergeron; Akshay Venkatesh The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu, Volume 12 (2013) no. 2, pp. 391-447 | DOI | MR | Zbl

[5] Glen E. Bredon Sheaf theory, Graduate Texts in Mathematics, 170, Springer, 1997, xii+502 pages | DOI | MR | Zbl

[6] Jeffrey F. Brock; Nathan M. Dunfield Injectivity radii of hyperbolic integer homology 3-spheres, Geom. Topol., Volume 19 (2015) no. 1, pp. 497-523 | DOI | MR | Zbl

[7] Frank Calegari; Akshay Venkatesh A torsion Jacquet–Langlands correspondence, Astérisque, 409, Société Mathématique de France, 2019 | Zbl

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

[9] Michael Farber Geometry of growth: approximation theorems for $L^{2}$ invariants, Math. Ann., Volume 311 (1998) no. 2, pp. 335-375 | DOI | MR | Zbl

[10] Tobias Finis; Erez Lapid An approximation principle for congruence subgroups, J. Eur. Math. Soc., Volume 20 (2018) no. 5, pp. 1075-1138 | DOI | MR | Zbl

[11] Tobias Finis; Erez Lapid An approximation principle for congruence subgroups. II: application to the limit multiplicity problem, Math. Z., Volume 289 (2018) no. 3-4, 1357.1380 pages | MR | Zbl

[12] Tobias Finis; Erez Lapid; Werner Müller Limit multiplicities for principal congruence subgroups of $GL (n)$ and $SL (n)$ , J. Inst. Math. Jussieu, Volume 14 (2015) no. 3, pp. 589-638 | DOI | MR | Zbl

[13] Mikolaj Fraczyk Strong Limit Multiplicity for arithmetic hyperbolic surfaces and $3$ -manifolds (2016) (https://arxiv.org/abs/1612.05354)

[14] David L. de George; Nolan R. Wallach Limit formulas for multiplicities in $L^{2} (Γ ∖ G)$ , Ann. Math., Volume 107 (1978) no. 1, pp. 133-150 | DOI | MR | Zbl

[15] Roger Godement The decomposition of $L^{2} (G / Γ)$ for $Γ = SL (2, Z)$ , Algebraic groups and discontinuous subgroups (Colorado Boulder, 1965) (Proceedings of Symposia in Pure Mathematics), Volume 9, American Mathematical Society, 1966, pp. 211-224 | MR | Zbl

[16] Daniel Gorenstein Finite groups, Chelsea Publishing, 1980, xvii+519 pages | MR | Zbl

[17] Paul E. Gunnells; Dan Yasaki Modular forms and elliptic curves over the cubic field of discriminant $- 23$ , Int. J. Number Theory, Volume 9 (2013) no. 1, pp. 53-76 | DOI | MR | Zbl

[18] Günter Harder Eisenstein cohomology of arithmetic groups. The case ${GL}_{2}$ , Invent. Math., Volume 89 (1987) no. 1, pp. 37-118 | DOI | MR | Zbl

[19] Thang T. Q. Lê Growth of homology torsion in finite coverings and hyperbolic volume, Ann. Inst. Fourier, Volume 68 (2018) no. 2, pp. 611-645 | MR | Zbl

[20] Michael Lipnowski; Mark Stern Geometry of the smallest 1-form Laplacian eigenvalue on hyperbolic manifolds, Geom. Funct. Anal., Volume 28 (2018) no. 6, pp. 1717-1755 | DOI | MR | Zbl

[21] Wolfgang Lück Approximating $L^{2}$ -invariants by their finite-dimensional analogues, Geom. Funct. Anal., Volume 4 (1994) no. 4, pp. 455-481 | DOI | MR | Zbl

[22] Wolfgang Lück $L^{2}$ -invariants: theory and applications to geometry and $K$ -theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 44, Springer, 2002, xvi+595 pages | MR | Zbl

[23] Pere Menal-Ferrer; Joan Porti Twisted cohomology for hyperbolic three manifolds, Osaka J. Math., Volume 49 (2012) no. 3, pp. 741-769 | MR | Zbl

[24] Jinsung Park Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps, J. Funct. Anal., Volume 257 (2009) no. 6, pp. 1713-1758 | DOI | MR | Zbl

[25] Jonathan Pfaff Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups, Ann. Global Anal. Geom., Volume 45 (2014) no. 4, pp. 267-285 | DOI | MR | Zbl

[26] Jean Raimbault Asymptotics of analytic torsion for hyperbolic three–manifolds (2012) (to appear in Comment. Math. Helv., https://arxiv.org/abs/1212.3161) | Zbl

[27] Jean Raimbault Torsion homologique dans les revêtements finis, Université Pierre et Marie Curie (France) (2012) (Ph. D. Thesis) | Zbl

[28] Jean Raimbault Analytic, Reidemeister and homological torsion for congruence three–manifolds (2013) (https://arxiv.org/abs/1307.2845v1)

[29] Jean Raimbault On the convergence of arithmetic orbifolds, Ann. Inst. Fourier, Volume 67 (2017) no. 6, pp. 2547-2596 | DOI | MR | Zbl

[30] Daniel B. Ray; Isadore M. Singer $R$ -torsion and the Laplacian on Riemannian manifolds, Adv. Math., Volume 7 (1971), pp. 145-210 | MR | Zbl

[31] Gordan Savin Limit multiplicities of cusp forms, Invent. Math., Volume 95 (1989) no. 1, pp. 149-159 | DOI | MR | Zbl

[32] Peter Scholze On torsion in the cohomology of locally symmetric varieties, Ann. Math., Volume 182 (2015) no. 3, pp. 945-1066 | DOI | MR | Zbl

[33] Mehmet Haluk Şengün On the integral cohomology of Bianchi groups, Exp. Math., Volume 20 (2011) no. 4, pp. 487-505 | DOI | MR | Zbl

[34] Jean-Pierre Serre Le problème des groupes de congruence pour SL2, Ann. Math., Volume 92 (1970), pp. 489-527 | DOI | MR | Zbl

Cité par Sources :