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Arithmetic rigidity and units in group rings


Author: F. E. A. Johnson
Journal: Trans. Amer. Math. Soc. 353 (2001), 4623-4635
MSC (2000): Primary 20C05; Secondary 20G20, 22E40
DOI: https://doi.org/10.1090/S0002-9947-01-02816-1
Published electronically: May 9, 2001
MathSciNet review: 1851185
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Abstract:

For any finite group $G$ 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 $G$ the converse is exactly true.


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Additional Information

F. E. A. Johnson
Affiliation: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
Email: feaj@math.ucl.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-01-02816-1
Received by editor(s): November 12, 1999
Received by editor(s) in revised form: August 28, 2000
Published electronically: May 9, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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