Arithmetic rigidity and units in group rings

Johnson, F.

doi:10.1090/S0002-9947-01-02816-1

Arithmetic rigidity and units in group rings
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by F. E. A. Johnson

Trans. Amer. Math. Soc. 353 (2001), 4623-4635

DOI: https://doi.org/10.1090/S0002-9947-01-02816-1

Published electronically: May 9, 2001

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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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