Transactions of the American Mathematical Society

Author(s): F. E. A. Johnson
Journal: Trans. Amer. Math. Soc. 353 (2001), 4623-4635.
MSC (2000): Primary 20C05; Secondary 20G20, 22E40
Posted: May 9, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

For any finite group

the group $U(\mathbf{Z}[G])$ of units in the integral group ring $\mathbf{Z}[G]$ is an arithmetic group in a reductive algebraic group, namely the Zariski closure of $\mathbf{SL}_1(\mathbf{Q}[G])$ . In particular, the isomorphism type of the $\mathbf{Q}$ -algebra $\mathbf{Q}[G]$ determines the commensurability class of $U(\mathbf{Z}[G])$ ; we show that, to a large extent, the converse is true. In fact, subject to a certain restriction on the $\mathbf{Q}$ -representations of

the converse is exactly true.

1.: A. A. Albert; Involutorial simple algebras and real Riemann matrices: Ann. of Math. 36 (1935) 886 - 964.
2.: S.A. Amitsur; Finite subgroups of division rings: Trans. Amer. Math. Soc. 80 (1955) 361 - 386. MR 17:577c
3.: R. Bieri and B. Eckmann; Groups with homological duality generalising Poincaré Duality: Invent. Math. 20 (1973) 103 - 124. MR 49:5204
4.: A. Borel; Density and maximality of arithmetic subgroups: J. Reine Angew. Math. 224 (1966) 78-89. MR 34:5824
5.: A. Borel; Linear algebraic groups: Benjamin, New York 1969. MR 40:4273
6.: A. Borel; Introduction aux groupes arithmétiques: Hermann, Paris 1969. MR 39:5577
7.: A. Borel and Harish-Chandra; Arithmetic subgroups of algebraic groups: Ann. of Math. 65 (1962) 485-535. MR 26:5081
8.: A. Borel and J.P. Serre; Corners and arithmetic groups: Comment. Math. Helv. 48 (1973) 436-491. MR 52:8337
9.: K.L. Fields; On the Brauer-Speiser Theorem: Bull. A.M.S. 77 (1971) 223. MR 42:3192
10.: F.E.A. Johnson; On the existence of irreducible discrete subgroups in isotypic Lie groups of classical type: Proc. L.M.S. 56 (1988) 51 - 77. MR 89f:22017
11.: F.E.A. Johnson; On the uniqueness of arithmetic structures: Proc. Roy. Soc. Edin. 124A (1994) 1037 - 1044. MR 95h:22013
12.: F.E.A. Johnson; Linear properties of poly-Fuchsian groups: Collectanea Math. 45, 2 (1994) 183-203. MR 95m:20030
13.: S.P. Kerckhoff; The Nielsen realisation problem: Ann. of Math. 117 (1983) 235 - 265. MR 85e:32029
14.: G. D. Mostow; Strong rigidity of locally symmetric spaces: Annals of Mathematics Studies no. 78. Princeton University Press 1974. MR 52:5874
15.: G. Margulis; Discrete subgroups of semisimple Lie groups: Ergebnisse der Math. 3 Folge, Bd. 17. Springer-Verlag, Berlin (1991). MR 92h:22021
16.: V. Platonov and A. Rapinchuk: Algebraic groups and number theory. Academic Press (1994). MR 95b:11039
17.: M.S. Raghunathan; Discrete subgroups of Lie groups: Ergebnisse der Math. no. 68. Springer-Verlag (1972). MR 58:22394a
18.: J.P. Serre; Cohomologie Galoisienne: Springer Lecture Notes 5 (1965). MR 34:1328
19.: G. Shimura; On analytic families of polarised abelian varieties and automorphic function: Ann. of Math. 78 (1963) 149 - 192. MR 27:5934
20.: J.R. Stallings; On torsion free groups with infinitely many ends: Ann. of Math. 88 (1968) 312-334. MR 37:4153

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20C05, 20G20, 22E40

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

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS