Describing units of integral group rings up to commensurability

Eisele, Florian; Kiefer, Ann; Gelder, Inneke Van

doi:10.1016/j.jpaa.2014.09.031

Cited by 14 publications

(23 citation statements)

References 24 publications

(51 reference statements)

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“…We hence embed SL 2 (( x,y Z )) into SL 2 (( −1,−1 R )) and re-use the same technique. Note that in [10], the authors showed that the exceptional components of type 3 can be reduced to the following three matrix rings over quaternion algebras:…”

Section: Introductionmentioning

confidence: 99%

“…This makes it possible to describe generators of SL 2 (O) with O a maximal order in these algebras by imitating the classical way, based on the Euclidean algorithm, in which generators for SL 2 (Z) are computed. For details see [10,Proposition 4.1]. So for the pure purpose of units in group rings, there is a way of computing generators for the unit group of a maximal order in components of type 3 and the details may be found in [10].…”

Section: Introductionmentioning

confidence: 99%

“…In [6], the authors apply Voronoï's algorithm, while in [20], the authors consider actions on direct products of hyperbolic spaces. The exceptional components of type 3 are until now barely dealt with, except for the paper [10], which we discuss later. The goal of this paper is to develop a new method that deals with these components.…”

Section: Introductionmentioning

confidence: 99%

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On units in orders in 2-by-2 matrices over quaternion algebras with rational center

Kiefer

2020

Groups Geom. Dyn.

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We generalize an algorithm established in earlier work [19] to compute finitely many generators for a subgroup of finite index of a group acting discontinuously on hyperbolic space of dimension 2 and 3, to hyperbolic space of higher dimensions using Clifford algebras. We hence get an algorithm which gives a finite set of generators up to finite index of a discrete subgroup of Vahlen's group, i.e. a group of 2-by-2 matrices with entries in the Clifford algebra satisfying certain conditions. The motivation comes from units in integral group rings and this new algorithm allows to handle unit groups of orders in 2-by-2 matrices over rational quaternion algebras. The rings investigated are part of the so-called exceptional components of a rational group algebra.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On units in orders in 2-by-2 matrices over quaternion algebras with rational center

Kiefer

2020

Groups Geom. Dyn.

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“…Q ) are very limited. It was proven by Eisele, Kiefer and Van Gelder [14] that only a finite number of division algebras can occur and, moreover, the possible parameters have been described. Together with the results of Bächle, Janssens, Jespers, Kiefer and D. Temmerman in [2] one has the following result.…”

Section: Rational Wedderburn Decompositionmentioning

confidence: 99%

Structure of group rings and the group of units of integral group rings: an invitation

Jespers¹

2020

Preprint

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During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group U (ZG) of the integral group ring ZG of a finite group G. These constructions rely on explicit constructions of units in ZG and proofs of main results make use of the description of the Wedderburn components of the rational group algebra QG. The latter relies on explicit constructions of primitive central idempotents and the rational representations of G. It turns out that the existence of reduced two degree representations play a crucial role. Although the unit group is far from being understood, some structure results on this group have been obtained. In this paper we give a survey of some of the fundamental results and the essential needed techniques.

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“…For producing this table, the classifications of either Banieqbal or Nebe can be used. The finite subgroups having M 2 (Q) or M 2 (K) with K a quadratic imaginary extension of Q are shown to be a subgroup of GL(2, 25) in [EKVG15].…”

Section: Introductionmentioning

confidence: 99%

A classification of exceptional components in group algebras over abelian number fields

2016

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When considering the unit group of O F G (O F the ring of integers of an abelian number field F and a finite group G) certain components in the Wedderburn decomposition of F G cause problems for known generic constructions of units; these components are called exceptional. Exceptional components are divided into two types: type 1 are division rings, type 2 are 2 × 2-matrix rings. For exceptional components of type 1 we provide infinite classes of division rings by describing the seven cases of minimal groups (w.r.t. quotients) having those division rings in their Wedderburn decomposition over F . We also classify the exceptional components of type 2 appearing in group algebras of a finite group over number fields F by describing all 58 finite groups G having a faithful exceptional Wedderburn component of this type in F G.

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Describing units of integral group rings up to commensurability

Cited by 14 publications

References 24 publications

On units in orders in 2-by-2 matrices over quaternion algebras with rational center

On units in orders in 2-by-2 matrices over quaternion algebras with rational center

Structure of group rings and the group of units of integral group rings: an invitation

A classification of exceptional components in group algebras over abelian number fields

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