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


Authors: E. Jespers, S. O. Juriaans, A. Kiefer, A. de A. e Silva and A. C. Souza Filho
Journal: Math. Comp. 84 (2015), 1489-1520
MSC (2010): Primary 16S34, 16U60; Secondary 20C05
DOI: https://doi.org/10.1090/S0025-5718-2014-02865-2
Published electronically: December 30, 2014
MathSciNet review: 3315518
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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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Additional Information

E. Jespers
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
Email: efjesper@vub.ac.be

S. O. Juriaans
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo (IME-USP), Caixa Postal 66281, São Paulo, CEP 05315-970 - Brazil
Email: ostanley@usp.br

A. Kiefer
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
Email: akiefer@vub.ac.be

A. de A. e Silva
Affiliation: Universidade Federal da Paraíba, Centro de Ciências Exatas e da Natureza - Campus I, Departamento de Matemática. Cidade Universitária Castelo Branco III 58051-900 - Joao Pessoa, PB - Brazil
Email: andrade@mat.ufpb.br

A. C. Souza Filho
Affiliation: Escola de Artes, Ciências e Humanidades, Universidade de São Paulo (EACH-USP), Rua Arlindo Béttio, 1000, Ermelindo Matarazzo, São Paulo, CEP 03828-000 - Brazil
Email: acsouzafilho@usp.br

DOI: https://doi.org/10.1090/S0025-5718-2014-02865-2
Keywords: Units, group ring, fundamental domain, generators
Received by editor(s): December 17, 2012
Received by editor(s) in revised form: July 25, 2013
Published electronically: December 30, 2014
Additional Notes: The first author was supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Flanders).
The second author was partially supported by CNPq and FAPESP-Brazil, while visiting the Vrije Universiteit Brussel.
The third author was supported by Fonds voor Wetenschappelijk Onderzoek (Flanders)-Belgium.
The fourth author was supported by FAPESP and CNPq-Brazil.
The fifth author was supported by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo), Proc. 2008/57930-1 and 2011/11315-7.
Article copyright: © Copyright 2014 American Mathematical Society

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