From the Poincaré Theorem to generators of the unit group of integral group rings of finite groups

Jespers, E.; Juriaans, S.; Kiefer, A.; de A. e Silva, A.; Souza Filho, A.

doi:10.1090/S0025-5718-2014-02865-2

From the Poincaré Theorem to generators of the unit group of integral group rings of finite groups
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by E. Jespers, S. O. Juriaans, A. Kiefer, A. de A. e Silva and A. C. Souza Filho;

Math. Comp. 84 (2015), 1489-1520

DOI: https://doi.org/10.1090/S0025-5718-2014-02865-2

Published electronically: December 30, 2014

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Abstract:

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\mathbb {Z} G$ of a finite nilpotent group $G$, this provided the rational group algebra $\mathbb {Q} G$ does not have simple components that are division classical quaternion algebras or two-by-two matrices over a classical quaternion algebra with center $\mathbb {Q}$. The main difficulty is to deal with orders in quaternion algebras over the rationals or a quadratic imaginary extension of the rationals. In order to deal with these we give a finite and easy implementable algorithm to compute a polyhedron containing a fundamental domain in the hyperbolic three space $\mathbb {H}^3$ (respectively, hyperbolic two space $\mathbb {H}^2$) for a discrete subgroup of $\mathrm {PSL}_2(\mathbb {C})$ (respectively, $\mathrm {PSL}_2(\mathbb {R})$) of finite covolume. Our results on group rings are a continuation of earlier work of Ritter and Sehgal, Jespers and Leal.

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