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

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


	Document type :	Scientific journals : Article
	Discipline(s) :	Physical, chemical, mathematical & earth Sciences : Mathematics
	To cite this reference:	http://hdl.handle.net/10993/45733

Title :	On units in orders in 2-by-2 matrices over quaternion algebras with rational center
Language :	English
Author, co-author :	Kiefer, Ann [University of Luxembourg > Faculty of Humanities, Education and Social Sciences (FHSE) > LUCET]
Publication date :	2020
Journal title :	Groups, Geometry, and Dynamics
Volume :	14
Issue/season :	1
Pages :	213--242
Peer reviewed :	Yes
ISSN :	1661-7207
e-ISSN :	1661-7215
Keywords :	[en] Hyperbolic Geometry ; Presentation ; Clifford Algebra ; Quaternion Algebra ; Group Rings ; Unit Group
Abstract :	[en] We generalize an algorithm established in earlier work [21] to compute finitely many generators for a subgroup of finite index of an arithmetic group acting properly discon- tinuously 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 of finite index subgroups 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.
Permalink :	http://hdl.handle.net/10993/45733
DOI :	10.4171/ggd/541

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