Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

The homological torsion of PSL$ _2$ of the imaginary quadratic integers


Author: Alexander D. Rahm
Journal: Trans. Amer. Math. Soc. 365 (2013), 1603-1635
MSC (2010): Primary 11F75, 22E40, 57S30; Secondary 55N91, 19L47, 55R35
DOI: https://doi.org/10.1090/S0002-9947-2012-05690-X
Published electronically: August 7, 2012
MathSciNet review: 3003276
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Bianchi groups are the groups (P) $ \mathrm {SL_2}$ over a ring of integers in an imaginary quadratic number field. 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 expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups.

Furthermore, this correspondence facilitates the computation of the equivariant $ K$-homology of the Bianchi groups. By the Baum/Connes conjecture, which is satisfied by the Bianchi groups, we obtain the $ K$-theory of their reduced $ C^*$-algebras in terms of isomorphic images of their equivariant $ K$-homology.


References [Enhancements On Off] (What's this?)

  • [1] Alejandro Adem and R. James Milgram, Cohomology of finite groups, Second Edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 309, Springer-Verlag, Berlin, 2004. MR 2035696 (2004k:20109)
  • [2] Bill Allombert, Christian Batut, Karim Belabas, Dominique Bernardi, Henri Cohen, Francisco Diaz y Diaz, Yves Eichenlaub, Xavier Gourdon, Louis Granboulan, Bruno Haible, Guillaume Hanrot, Pascal Letard, Gerhard Niklasch, Michel Olivier, Thomas Papanikolaou, Xavier Roblot, Denis Simon, Emmanuel Tollis, Ilya Zakharevitch, and the PARI group, PARI/GP, version 2.4.3, specialized computer algebra system, Bordeaux, 2010, http://pari.math.u-bordeaux.fr/.
  • [3] Maria T. Aranés, Modular symbols over number fields, Ph.D. Thesis, 2010.
  • [4] Avner Ash, Deformation retracts with lowest possible dimension of arithmetic quotients of self-adjoint homogeneous cones, Math. Ann. 225 (1977), no. 1, 69-76. MR 0427490 (55 #522)
  • [5] Ethan Berkove, The mod-$ 2$ cohomology of the Bianchi groups, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4585-4602. MR 1675241 (2001b:11043)
  • [6] Luigi Bianchi, Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî, Math. Ann. 40 (1892), no. 3, 332-412. MR 1510727
  • [7] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag. MR 1744486 (2000k:53038)
  • [8] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, 1982. MR 672956 (83k:20002)
  • [9] Jeremy Bygott, Modular forms and modular symbols over imaginary quadratic fields, Ph.D. Thesis, University of Exeter, 1999, http://www.warwick.ac.uk/staff/J.E.Cremona/theses/bygott.pdf.
  • [10] J. E. Cremona and M. P. Lingham, Finding all elliptic curves with good reduction outside a given set of primes, Experiment. Math. 16 (2007), no. 3, 303-312. MR 2367320 (2008k:11057)
  • [11] J. E. Cremona and E. Whitley, Periods of cusp forms and elliptic curves over imaginary quadratic fields, Math. Comp. 62 (1994), no. 205, 407-429. MR 1185241 (94c:11046)
  • [12] J. E. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compositio Math. 51 (1984), no. 3, 275-324. MR 743014 (85j:11063)
  • [13] Graham Ellis, Homological algebra programming, Computational Group Theory and the Theory of Groups, 2008, pp. 63-74. MR 2478414 (2009k:20001)
  • [14] Jürgen Elstrodt, Fritz Grunewald, and Jens Mennicke, Groups acting on hyperbolic space, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1483315 (98g:11058)
  • [15] Benjamin 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)
  • [16] Dieter Flöge, Zur Struktur der PSL$ \sb {2}$ über einigen imaginär-quadratischen Zahlringen, Math. Z. 183 (1983), no. 2, 255-279 (German). MR 704107 (85c:11043)
  • [17] Dieter Flöge, Dissertation: Zur Struktur der PSL$ \sb {2}$ über einigen imaginär-quadratischen Zahlringen, Ph.D. Thesis, 1980 (German).
  • [18] Paul E. Gunnells, Modular symbols for $ {\mathbb{Q}}$-rank one groups and Voronoĭreduction, J. Number Theory 75 (1999), no. 2, 198-219. MR 1681629 (2000c:11084)
  • [19] Georges Humbert, Sur la réduction des formes d'Hermite dans un corps quadratique imaginaire, C. R. Acad. Sci. Paris 16 (1915), 189-196 (French).
  • [20] Pierre Julg and Gennadi Kasparov, Operator $ K$-theory for the group $ {\rm SU}(n,1)$, J. Reine Angew. Math. 463 (1995), 99-152. MR 1332908 (96g:19006)
  • [21] Felix Klein, Ueber binäre Formen mit linearen Transformationen in sich selbst, Math. Ann. 9 (1875), no. 2, 183-208. MR 1509857
  • [22] Mark Lingham, Modular forms and elliptic curves over imaginary quadratic fields, Ph.D. Thesis, 2005.
  • [23] Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR 1937957 (2004i:57021)
  • [24] Eduardo R. Mendoza, Cohomology of PGL$ \sb {2}$ over imaginary quadratic integers, Bonner Mathematische Schriften [Bonn Mathematical Publications], 128, Dissertation, Rheinische Friedrich-Wilhelms-Universität, Mathematisches Institut. MR 611515 (82g:22012)
  • [25] Henri Poincaré, Mémoire : Les groupes kleinéens, Acta Math. 3 (1883), no. 1, 49-92. MR 1554613
  • [26] Alexander 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 group homology of the Bianchi groups.
  • [27] Alexander D. Rahm and Mathias Fuchs, The integral homology of $ {\rm PSL}_2$ of imaginary quadratic integers with nontrivial class group, J. Pure Appl. Algebra 215 (2011), no. 6, 1443-1472. MR 2769243
  • [28] Alexander D. Rahm, (Co)homologies and $ {K}$-theory of Bianchi groups using computational geometric models, Ph.D. Thesis, Institut Fourier, Université de Grenoble et Universität Göttingen, soutenue le 15 octobre 2010. http://tel.archives-ouvertes.fr/tel-00526976/.
  • [29] John G. Ratcliffe, Foundations of hyperbolic manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 149, Springer. MR 2249478 (2007d:57029)
  • [30] Rubén J. Sánchez-García, Bredon homology and equivariant $ {K}$-homology of $ {\rm {S}{L} }(3,{\mathbb{ {Z} }})$, J. Pure Appl. Algebra 212 (2008), no. 5, 1046-1059. MR 2387584 (2009b:19007)
  • [31] Joachim Schwermer and Karen Vogtmann, The integral homology of SL$ _{2}$ and PSL$ _{2}$ of Euclidean imaginary quadratic integers, Comment. Math. Helv. 58 (1983), no. 4, 573-598. MR 728453 (86d:11046), Zbl 0545.20031
  • [32] Mehmet H. Sengün, On the (co)homology of Bianchi groups, Exp. Math. 20 (2011), no. 4, 487-505. MR 2859903
  • [33] Richard G. Swan, Generators and relations for certain special linear groups, Advances in Math. 6 (1971), 1-77. MR 0284516 (44:1741)
  • [34] Karen Vogtmann, Rational homology of Bianchi groups, Math. Ann. 272 (1985), no. 3, 399-419. MR 799670 (87a:22025)
  • [35] C. Terence C. Wall, Resolutions for extensions of groups, Proc. Cambridge Philos. Soc. 57 (1961), 251-255. MR 0178046 (31 #2304)
  • [36] Dan Yasaki, Hyperbolic tessellations associated to Bianchi groups, Algorithmic Number Theory, Lecture Notes in Comput. Sci. 6197, 2010, pp. 385-396. MR 2721434

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11F75, 22E40, 57S30, 55N91, 19L47, 55R35

Retrieve articles in all journals with MSC (2010): 11F75, 22E40, 57S30, 55N91, 19L47, 55R35


Additional Information

Alexander D. Rahm
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
Address at time of publication: Department of Mathematics, National University of Ireland at Galway, University Road, Galway, Ireland
Email: Alexander.Rahm@Weizmann.ac.il, Alexander.Rahm@nuigalway.ie

DOI: https://doi.org/10.1090/S0002-9947-2012-05690-X
Received by editor(s): May 16, 2011
Received by editor(s) in revised form: August 13, 2011
Published electronically: August 7, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society