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.

We establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL 2 of the imaginary quadratic integers), in terms of elementary number-theoretic quantities. To achieve this, we introduce a novel technique in the computation of Bredon homology: representation ring splitting, which allows us to adapt the recent technique of torsion subcomplex reduction from group homology to Bredon homology.

Pour la K-homologie équivariante de tous les groupes de Bianchi (PSL 2 sur les entiers quadratiques imaginaires), nous démontrons des formules pour la partie due à la torsion, en termes de quantités élémentaires de la théorie des nombres. Pour arriver à cette fin, nous introduisons une nouvelle technique pour le calcul de l’homologie de Bredon : un scindage des anneaux de représentation, qui nous permet d’adapter la technique récente de réduction des sous-complexes de torsion, développée pour l’homologie des groupes, à notre usage pour l’homologie de Bredon.

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DOI: 10.5802/aif.3047
Classification: 55N91, 19L47
Keywords: Equivariant homology and cohomology, Equivariant $K$-theory, Bianchi groups, PSL$_2$ of the imaginary quadratic integers
Mot clés : Homologie et cohomologie équivariantes, $K$-théorie équivariante, Groupes de Bianchi, PSL$_2$ sur les entiers quadratiques imaginaires

Rahm, Alexander D. 1

1 Université du Luxembourg, Mathematics Research Unit 6, rue Coudenhove-Kalergi L-1359 Luxembourg-Kirchberg Luxembourg
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Rahm, Alexander D. 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. https://aif.centre-mersenne.org/articles/10.5802/aif.3047/

[1] Baum, Paul; Connes, Alain; Higson, Nigel Classifying space for proper actions and K-theory of group C * -algebras, C * -algebras: 1943–1993 (San Antonio, TX, 1993) (Contemp. Math.), Volume 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240-291 | DOI

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

[3] Brown, Kenneth S. Cohomology of groups, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994, x+306 pages (Corrected reprint of the 1982 original)

[4] Cherix, Pierre-Alain; Cowling, Michael; Jolissaint, Paul; Julg, Pierre; Valette, Alain Groups with the Haagerup property, Progress in Mathematics, 197, Birkhäuser Verlag, Basel, 2001, viii+126 pages (Gromov’s a-T-menability) | DOI

[5] Cherix, Pierre-Alain; Martin, Florian; Valette, Alain Spaces with measured walls, the Haagerup property and property (T), Ergodic Theory Dynam. Systems, Volume 24 (2004) no. 6, pp. 1895-1908 | DOI

[6] Flöge, Dieter Zur Struktur der PSL 2 über einigen imaginär-quadratischen Zahlringen, Math. Z., Volume 183 (1983) no. 2, pp. 255-279 | DOI

[7] Fuchs, Mathias Equivariant K-homology of Bianchi groups in the case of non-trivial class group (to appear in Münster Journal of Mathematics, http://wwwmath1.uni-muenster.de/mjm/acc/Fuchs.pdf)

[8] Harder, G. A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. École Norm. Sup. (4), Volume 4 (1971), pp. 409-455

[9] Higson, Nigel; Kasparov, Gennadi E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math., Volume 144 (2001) no. 1, pp. 23-74 | DOI

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

[11] Klein, Felix Ueber binäre Formen mit linearen Transformationen in sich selbst, Math. Ann., Volume 9 (1875) no. 2, pp. 183-208 | DOI

[12] Krämer, Norbert Die Konjugationsklassenanzahlen der endlichen Untergruppen in der Norm-Eins-Gruppe von Maximalordnungen in Quaternionenalgebren, Universität Bonn, Germany (1980) (Ph. D. Thesis)

[13] Krämer, Norbert Imaginärquadratische Einbettung von Maximalordnungen rationaler Quaternionenalgebren, und die nichtzyklischen endlichen Untergruppen der Bianchi-Gruppen (2015) (preprint, http://hal.archives-ouvertes.fr/hal-00720823/en/)

[14] Lafont, Jean-Francois; Ortiz, Ivonne; Rahm, Alexander D.; Sanchez-Garcia, Ruben Bredon homology and equivariant K-homology of hyperbolic Coxeter groups (work in advanced progress)

[15] Lück, Wolfgang; Oliver, Bob Chern characters for the equivariant K-theory of proper G-CW-complexes, Cohomological methods in homotopy theory (Bellaterra, 1998) (Progr. Math.), Volume 196, Birkhäuser, Basel, 2001, pp. 217-247

[16] Mendoza, Eduardo R. Cohomology of PGL 2 over imaginary quadratic integers, Bonner Mathematische Schriften [Bonn Mathematical Publications], 128, Universität Bonn, Mathematisches Institut, Bonn, 1979, vi+83 pages (Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1979)

[17] Mislin, Guido; Valette, Alain Proper group actions and the Baum-Connes conjecture, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2003, viii+131 pages | DOI

[18] Poincaré, H. Mémoire : Les groupes kleinéens, Acta Math., Volume 3 (1883) no. 1, pp. 49-92 | DOI

[19] Rahm, Alexander D. Bianchi.gp, 2010 Open source program (GNU general public license), validated by the CNRS: http://www.projet-plume.org/fiche/bianchigp, Part of the Pari/GP Development Center scripts library

[20] Rahm, Alexander D. (Co)homologies and K-theory of Bianchi groups using computational geometric models, Institut Fourier, Université de Grenoble, France and Universität Göttingen, Germany (2010) (Ph. D. Thesis)

[21] Rahm, Alexander D. Homology and K-theory of the Bianchi groups, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 11-12, pp. 615-619 | DOI

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

[23] Rahm, Alexander D. The homological torsion of PSL 2 of the imaginary quadratic integers, Trans. Amer. Math. Soc., Volume 365 (2013) no. 3, pp. 1603-1635 | DOI

[24] Rahm, Alexander D. Accessing the cohomology of discrete groups above their virtual cohomological dimension, J. Algebra, Volume 404 (2014), pp. 152-175 | DOI

[25] Rahm, Alexander D.; Fuchs, Mathias The integral homology of PSL 2 of imaginary quadratic integers with nontrivial class group, J. Pure Appl. Algebra, Volume 215 (2011) no. 6, pp. 1443-1472 | DOI

[26] Sánchez-García, Rubén Bredon homology and equivariant K-homology of SL (3,), J. Pure Appl. Algebra, Volume 212 (2008) no. 5, pp. 1046-1059 | DOI

[27] Sánchez-García, Rubén J. Equivariant K-homology for some Coxeter groups, J. Lond. Math. Soc. (2), Volume 75 (2007) no. 3, pp. 773-790 | DOI

[28] Serre, Jean-Pierre Le problème des groupes de congruence pour SL2, Ann. of Math. (2), Volume 92 (1970), pp. 489-527

[29] Serre, Jean-Pierre Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977, x+170 pages (Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42)

[30] Słomińska, J. On the equivariant Chern homomorphism, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., Volume 24 (1976) no. 10, pp. 909-913

[31] Soulé, Christophe The cohomology of SL 3 (Z), Topology, Volume 17 (1978) no. 1, pp. 1-22

[32] Valette, Alain Introduction to the Baum-Connes conjecture, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002, x+104 pages (From notes taken by Indira Chatterji, With an appendix by Guido Mislin) | DOI

[33] Vogtmann, Karen Rational homology of Bianchi groups, Math. Ann., Volume 272 (1985) no. 3, pp. 399-419 | DOI

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