Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

The Mod-2 cohomology of the Bianchi groups

Author(s): Ethan Berkove
Journal: Trans. Amer. Math. Soc. 352 (2000), 4585-4602.
MSC (2000): Primary 20J06; Secondary 11F75, 22E40
Posted: April 21, 2000
MathSciNet review: 1675241
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract:

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:

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 -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 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 Over Imaginary Quadratic Integers, Bonner Math. Schriften, vol. 128, 1980. MR 82g:22012
10.: J. Schwermer and K. Vogtmann, The Integral Homology of and 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

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|