Homology and <i>K</i>-theory of the Bianchi groups

Alexander D. Rahm

doi:10.1016/j.crma.2011.05.014

Homological Algebra/Group Theory

Homology and K-theory of the Bianchi groups
[Homologie et K-théorie des groupes de Bianchi]

Présenté par : Christophe Soulé

Alexander D. Rahm ¹

¹ Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 615-619.

Abstract (VO)
Résumé

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant K-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the K-theory of their reduced $C^{⁎}$ -algebras in terms of isomorphic images of the computed K-homology. We further find an application to Chen/Ruan orbifold cohomology.

Nous mettons en évidence une correspondance entre la torsion homologique des groupes de Bianchi et de nouveaux invariants géométriques, calculables grâce à leur action sur lʼespace hyperbolique. Nous lʼutilisons pour calculer explicitement leur homologie de groupe à coefficients entiers et leur K-homologie équivariante. En conséquence de la conjecture de Baum/Connes, qui est vérifiée pour ces groupes, nous obtenons la K-théorie de leurs C*-algèbres réduites en termes dʼimages isomorphes de la K-homologie calculée. Nous trouvons dʼailleurs une application à la cohomologie dʼorbi-espace de Chen/Ruan.

Reçu le : 2011-03-04
Accepté le : 2011-05-30
Publié le : 2011-06-01

DOI : 10.1016/j.crma.2011.05.014

Affiliations des auteurs :

Alexander D. Rahm ¹

¹ Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

@article{CRMATH_2011__349_11-12_615_0,
     author = {Alexander D. Rahm},
     title = {Homology and {\protect\emph{K}-theory} of the {Bianchi} groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {615--619},
     publisher = {Elsevier},
     volume = {349},
     number = {11-12},
     year = {2011},
     doi = {10.1016/j.crma.2011.05.014},
     language = {en},
}

TY  - JOUR
AU  - Alexander D. Rahm
TI  - Homology and K-theory of the Bianchi groups
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 615
EP  - 619
VL  - 349
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2011.05.014
LA  - en
ID  - CRMATH_2011__349_11-12_615_0
ER  -

%0 Journal Article
%A Alexander D. Rahm
%T Homology and K-theory of the Bianchi groups
%J Comptes Rendus. Mathématique
%D 2011
%P 615-619
%V 349
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2011.05.014
%G en
%F CRMATH_2011__349_11-12_615_0

Alexander D. Rahm. Homology and K-theory of the Bianchi groups. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 615-619. doi : 10.1016/j.crma.2011.05.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.05.014/

Bibliographie
Cité par

[1] R. Aurich, F. Steiner, H. Then, Numerical computation of Maass waveforms and an application to cosmology, Contribution to the Proceedings of the “International School on Mathematical Aspects of Quantum Chaos II”, Lecture Notes in Physics, Springer, 2004, in press.

[2] L. Bianchi Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî, Math. Ann., Volume 40 (1892) no. 3, pp. 332-412 (MR 1510727)

[3] K.S. Brown Cohomology of Groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, 1982

[4] W. Chen; Y. Ruan A new cohomology theory of orbifold, Comm. Math. Phys., Volume 248 (2004) no. 1, pp. 1-31

[5] J. Elstrodt; F. Grunewald; J. Mennicke Groups Acting on Hyperbolic Space, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998 MR 1483315 (98g:11058)

[6] B. Fine Algebraic Theory of the Bianchi Groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 129, Marcel Dekker Inc., New York, 1989 MR 1010229 (90h:20002)

[7] P. Julg; G. Kasparov Operator K-theory for the group $SU (n, 1)$ , J. Reine Angew. Math., Volume 463 (1995), pp. 99-152

[8] C. Maclachlan; A.W. Reid The Arithmetic of Hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003 MR 1937957 (2004i:57021)

[9] H. Poincaré Les groupes kleinéens, Mémoire, Acta Math., Volume 3 (1966) no. 1, pp. 49-92 (MR 1554613)

[10] A.D. Rahm, Bianchi.gp, Open source program (GNU general public license), 2010. Available at http://tel.archives-ouvertes.fr/tel-00526976/ this program computes a fundamental domain for the Bianchi groups in hyperbolic 3-space, the associated quotient space and essential information about the integral homology of the Bianchi groups.

[11] A.D. Rahm, (Co)homologies and K-theory of Bianchi groups using computational geometric models, PhD thesis, Institut Fourier, Université de Grenoble et Universität Göttingen, soutenue le 15 octobre 2010, http://tel.archives-ouvertes.fr/tel-00526976/.

[12] A.D. Rahm; M. Fuchs The integral homology of ${PSL}_{2}$ of imaginary quadratic integers with non-trivial class group, J. Pure Appl. Algebra, Volume 215 (2011), pp. 1443-1472

[13] J. Schwermer; K. Vogtmann The integral homology of ${SL}_{2}$ and ${PSL}_{2}$ of Euclidean imaginary quadratic integers, Comment. Math. Helv., Volume 58 (1983) no. 4, pp. 573-598

Sam Hughes On the equivariant $K$ - and $K O$ -homology of some special linear groups, Algebraic Geometric Topology, Volume 21 (2021) no. 7, pp. 3483-3512 | DOI:10.2140/agt.2021.21.3483 | Zbl:1509.19010
Ethan Berkove; Grant S. Lakeland; Alexander D. Rahm The mod 2 cohomology rings of congruence subgroups in the Bianchi groups, Journal of Algebraic Combinatorics, Volume 52 (2020) no. 4, pp. 527-560 | DOI:10.1007/s10801-019-00912-8 | Zbl:1472.11164
Bram Mesland; Mehmet Haluk Şengün Hecke operators in $K K$ -theory and the $K$ -homology of Bianchi groups, Journal of Noncommutative Geometry, Volume 14 (2020) no. 1, pp. 125-189 | DOI:10.4171/jncg/361 | Zbl:1473.11112
Awais Yousaf; Hanan Alolaiyan; Abdul Razaq; Muhammad Younis Evolution of ambiguous numbers under the actions of a Bianchi group, Journal of Taibah University for Science, Volume 14 (2020) no. 1, p. 615 | DOI:10.1080/16583655.2020.1760511
Alexander D. Rahm; Panagiotis Tsaknias Genuine Bianchi modular forms of higher level at varying weight and discriminant, Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 27-48 | DOI:10.5802/jtnb.1067 | Zbl:1466.11027
Alexander D. Rahm On the equivariant $K$ -homology of ${PSL}_{2}$ of the imaginary quadratic integers, Annales de l'Institut Fourier, Volume 66 (2016) no. 4, pp. 1667-1689 | DOI:10.5802/aif.3047 | Zbl:1360.55007
Ethan Berkove; Alexander D. Rahm The mod 2 cohomology rings of ${SL}_{2}$ of the imaginary quadratic integers. With an appendix by Aurel Page, Journal of Pure and Applied Algebra, Volume 220 (2016) no. 3, pp. 944-975 | DOI:10.1016/j.jpaa.2015.08.002 | Zbl:1401.11100
Mehmet Haluk Şengün Arithmetic aspects of Bianchi groups, Computations with modular forms. Proceedings of a summer school and conference, Heidelberg, Germany, August–September 2011, Cham: Springer, 2014, pp. 279-315 | DOI:10.1007/978-3-319-03847-6_11 | Zbl:1375.11039
Kevin Hutchinson A refined Bloch group and the third homology of ${SL}_{2}$ of a field, Journal of Pure and Applied Algebra, Volume 217 (2013) no. 11, pp. 2003-2035 | DOI:10.1016/j.jpaa.2013.01.001 | Zbl:1281.19003
Alexander D. Rahm; Mehmet Haluk Şengün On level one cuspidal Bianchi modular forms, LMS Journal of Computation and Mathematics, Volume 16 (2013), pp. 187-199 | DOI:10.1112/s1461157013000053 | Zbl:1294.11062
Alexander D. Rahm Higher torsion in the abelianization of the full Bianchi groups, LMS Journal of Computation and Mathematics, Volume 16 (2013), pp. 344-365 | DOI:10.1112/s1461157013000168 | Zbl:1328.11057
Alexander D. Rahm The homological torsion of ${PSL}_{2}$ of the imaginary quadratic integers, Transactions of the American Mathematical Society, Volume 365 (2013) no. 3, pp. 1603-1635 | DOI:10.1090/s0002-9947-2012-05690-x | Zbl:1307.11065

Cité par 12 documents. Sources : Crossref, zbMATH

Commentaires - Politique