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)



The Mod-2 cohomology of the Bianchi groups

Author: Ethan Berkove
Journal: Trans. Amer. Math. Soc. 352 (2000), 4585-4602
MSC (2000): Primary 20J06; Secondary 11F75, 22E40
Published electronically: April 21, 2000
MathSciNet review: 1675241
Full-text PDF

Abstract | References | Similar Articles | Additional Information


The Bianchi groups are a family of discrete subgroups of $PSL_2(\mathbb C)$which have group theoretic descriptions as amalgamated products and HNN extensions. Using Bass-Serre theory, we show how the cohomology of these two constructions relates to the cohomology of their pieces. We then apply these results to calculate the mod-2 cohomology ring for various Bianchi groups.

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

  • 1. A. Adem and J. Milgram, Cohomology of Finite Groups, Springer Verlag, 1994. MR 96f:20082
  • 2. R. Alperin, Homology of $SL_2({\mathbb Z}[\omega])$, Comment. Math. Helvetici 55 (1980), 364-377. MR 82f:20070
  • 3. A. Beardon, The Geometry of Discrete Groups, 2nd edition, Springer Verlag, 1995. MR 97d:22011
  • 4. E. Berkove and D. Juan-Pineda, On the $K$-theory of Bianchi Groups, Bol. Soc. Mat. Mexicana 2 (1996), 12-29. MR 97d:19003
  • 5. K. Brown, Cohomology of Groups, Springer Verlag, 1994. MR 96a:22072
  • 6. B. Fine, Algebraic Theory of the Bianchi Groups, Marcel Dekker, Inc. (Monographs and textbooks in pure and applied mathematics; 129), 1989. MR 90h:20002
  • 7. D. Flöge, Zur Struktur der $PSL_2$ über einigen imaginär-quadratischen Zahlringen, Math. Zeitschrift 183 (1983), 255-279. MR 85c:11043
  • 8. F. Grunewald and J. Schwermer, Subgroups of Bianchi Groups and Arithmetic Quotients of Hyperbolic 3-Space, Trans. of the AMS 335 (1993), 47-78. MR 93c:11024
  • 9. E. Mendoza, Cohomology of $PGL_2$ Over Imaginary Quadratic Integers, Bonner Math. Schriften, vol. 128, 1980. MR 82g:22012
  • 10. J. Schwermer and K. Vogtmann, The Integral Homology of $SL_2$ and $PSL_2$ of Euclidean Imaginary Quadratic Integers, Comment. Math. Helvetica 58 (1983), 573-598. MR 86d:11046
  • 11. J. P. Serre, Trees, Springer-Verlag, 1980. MR 82c:20083
  • 12. E. Spanier, Algebraic Topology, 1st reprinted version, Springer-Verlag, 1989. MR 96a:55001
  • 13. R. Swan, Generators and Relations for Certain Special Linear Groups, Adv. in Mathematics 6 (1971), 1-77. MR 44:1741
  • 14. K. Vogtmann, Rational Cohomology of Bianchi Groups, Math. Ann. 272 (1985), 399-419. MR 87a:22025

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20J06, 11F75, 22E40

Retrieve articles in all journals with MSC (2000): 20J06, 11F75, 22E40

Additional Information

Ethan Berkove
Affiliation: Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042

Received by editor(s): April 6, 1998
Published electronically: April 21, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society