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Presentations of groups acting discontinuously on direct products of hyperbolic spaces


Authors: E. Jespers, A. Kiefer and Á. del Río
Journal: Math. Comp. 85 (2016), 2515-2552
MSC (2010): Primary 20G20, 22E40, 16S34, 16U60
Published electronically: December 31, 2015
MathSciNet review: 3511291
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Abstract: The problem of describing the group of units $ \mathcal {U}(\mathbb{Z} G)$ of the integral group ring $ \mathbb{Z} G$ of a finite group $ G$ has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order $ \mathcal {O}$ in the simple epimorphic images $ A$ of the rational group algebra $ \mathbb{Q} G$. Making use of the presentation part of Poincaré's polyhedron theorem, Pita, del Río and Ruiz proposed such a method for a large family of finite groups $ G$ and consequently Jespers, Pita, del Río, Ruiz and Zalesskii described the structure of $ \mathcal {U}(\mathbb{Z} G)$ for a large family of finite groups $ G$. In order to handle many more groups, one would like to extend Poincaré's method to discontinuous subgroups of the group of isometries of a direct product of hyperbolic spaces. If the algebra $ A$ has degree 2 then via the Galois embeddings of the centre of the algebra $ A$ one considers the group of reduced norm one elements of the order $ \mathcal {O}$ as such a group and thus one would obtain a solution to the mentioned problem. This would provide presentations of the unit group of orders in the simple components of degree 2 of $ \mathbb{Q} G$ and in particular describe the unit group of $ \mathbb{Z} G$ for every group $ G$ with irreducible character degrees less than or equal to 2. The aim of this paper is to initiate this approach by executing this method on the Hilbert modular group, i.e., the projective linear group of degree two over the ring of integers in a real quadratic extension of the rationals. This group acts discontinuously on a direct product of two hyperbolic spaces of dimension two. The fundamental domain constructed is an analogue of the Ford domain of a Fuchsian or a Kleinian group.


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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

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

Á. del Río
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain
Email: adelrio@um.es

DOI: https://doi.org/10.1090/mcom/3071
Keywords: Hilbert modular group, discontinuous action on direct product of hyperbolic $2$-space, presentation, fundamental domain, group rings, unit group
Received by editor(s): January 7, 2015
Received by editor(s) in revised form: April 12, 2015
Published electronically: December 31, 2015
Additional Notes: The first author is supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Flanders). The second author is supported by Fonds voor Wetenschappelijk Onderzoek (Flanders)-Belgium. The third author is partially supported by the Spanish Government under Grant MTM2012-35240 with “Fondos FEDER”.
Article copyright: © Copyright 2015 American Mathematical Society