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.

Résumé
Abstract

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.

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.

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

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

On a question of Serre

Alexander D. Rahm

C. R. Math (2012)

Corrigendum to “On a question of Serre” [C. R. Acad. Sci. Paris, Ser. I 350 (2012) 741–744]

Alexander D. Rahm

C. R. Math (2019)

Bianchi–Euler system for relativistic fluids and Bel–Robinson type energy

Yvonne Choquet-Bruhat; James W. York

C. R. Math (2002)