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Abelianization and fixed point properties of units in integral group rings ; ; et al in accepted in Mathematische Nachrichten (2021) Detailed reference viewed: 20 (2 UL)On units in orders in 2-by-2 matrices over quaternion algebras with rational center Kiefer, Ann in Groups Geom. Dyn. (2020), 14(1), 213--242 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 15 (1 UL)Dirichlet-Ford domains and double Dirichlet domains ; ; Kiefer, Ann et al in Bull. Belg. Math. Soc. Simon Stevin (2016), 23(3), 465--479 We continue investigations started by Lakeland on Fuchsian and Kleinian groups which have a Dirichlet fundamental domain that also is a Ford domain in the upper half-space model of hyperbolic 2- and 3 ... [more ▼] We continue investigations started by Lakeland on Fuchsian and Kleinian groups which have a Dirichlet fundamental domain that also is a Ford domain in the upper half-space model of hyperbolic 2- and 3-space, or which have a Dirichlet domain with multiple centers. Such domains are called DF-domains and Double Dirichlet domains respectively. Making use of earlier obtained concrete formulas for the bisectors defining the Dirichlet domain of center i ∈ H 2 or center j ∈ H 3 , we obtain a simple condition on the matrix entries of the side- pairing transformations of the fundamental domain of a Fuchsian or Kleinian group to be a DF-domain. Using the same methods, we also complement a result of Lakeland stating that a cofinite Fuchsian group has a DF domain (or a Dirichlet domain with multiple centers) if and only if it is an index 2 subgroup of the discrete group G of reflections in a hyperbolic polygon. [less ▲] Detailed reference viewed: 20 (0 UL)Presentations of groups acting discontinuously on direct products of hyperbolic spaces ; Kiefer, Ann ; in Math. Comp. (2016), 85(301), 2515--2552 The problem of describing the group of units U(ZG) of the integral group ring ZG of a finite group G has attracted a lot of attention and providing presentations for such groups is a fundamental problem ... [more ▼] The problem of describing the group of units U(ZG) of the integral group ring ZG 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 O in the simple epimorphic images A of the rational group algebra QG. 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 U(ZG) 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 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 QG and in particular describe the unit group of ZG 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. [less ▲] Detailed reference viewed: 14 (0 UL)Revisiting Poincaré's Theorem on presentations of discontinuous groups via fundamental polyhedra ; Kiefer, Ann ; in Expo. Math. (2015), 33(4), 401--430 We give a new self-contained proof of Poincaré's Polyhedron Theorem on presentations of discon- tinuous groups of isometries of a Riemann manifold of constant curvature. The proof is not based on the ... [more ▼] We give a new self-contained proof of Poincaré's Polyhedron Theorem on presentations of discon- tinuous groups of isometries of a Riemann manifold of constant curvature. The proof is not based on the theory of covering spaces, but only makes use of basic geometric concepts. In a sense one hence obtains a proof that is of a more constructive nature than most known proofs. [less ▲] Detailed reference viewed: 27 (0 UL)Describing units of integral group rings up to commensurability ; Kiefer, Ann ; in J. Pure Appl. Algebra (2015), 219(7), 2901--2916 We restrict the types of 2 × 2-matrix rings which can occur as simple components in the Wedderburn decomposition of the rational group algebra of a finite group. This results in a description up to ... [more ▼] We restrict the types of 2 × 2-matrix rings which can occur as simple components in the Wedderburn decomposition of the rational group algebra of a finite group. This results in a description up to commensurability of the group of units of the integral group ring ZG for all finite groups G that do not have a non-commutative Frobenius complement as a quotient. [less ▲] Detailed reference viewed: 9 (0 UL)From the Poincaré theorem to generators of the unit group of integral group rings of finite groups ; ; Kiefer, Ann et al in Math. Comp. (2015), 84(293), 1489--1520 We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring ZG of a finite nilpotent group G, this provided the rational group ... [more ▼] We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring ZG of a finite nilpotent group G, this provided the rational group algebra QG does not have simple components that are division classical quaternion algebras or two-by-two matrices over a classical quaternion algebra with centre 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 fundamental domain in the hyperbolic three space H 3 (respectively hyperbolic two space H 2 ) for a discrete subgroup of PSL 2 (C) (respectively PSL 2 (R)) of finite covolume. Our results on group rings are a continuation of earlier work of Ritter and Sehgal, Jespers and Leal. [less ▲] Detailed reference viewed: 41 (0 UL)On pairs of commuting involutions in $ Sym(n)$ and $ Alt(n)$ Kiefer, Ann ; in Comm. Algebra (2013), 41(12), 4408--4418 The number of pairs of commuting involutions in Sym(n) and Alt(n) is determined up to isomorphism. It is also proven that, up to isomor- phism and duality, there are exactly two abstract regular polyhedra ... [more ▼] The number of pairs of commuting involutions in Sym(n) and Alt(n) is determined up to isomorphism. It is also proven that, up to isomor- phism and duality, there are exactly two abstract regular polyhedra on which the group Sym(6) acts as a regular automorphism group. [less ▲] Detailed reference viewed: 21 (0 UL)On the number of abstract regular polytopes whose automorphism group is a Suzuki simple group $ Sz(q)$ Kiefer, Ann ; in J. Combin. Theory Ser. A (2010), 117(8), 1248--1257 We determine, up to isomorphism and duality, the number of abstract regular polytopes of rank three, whose automorphism group is a Suzuki simple group Sz(q), with q an odd power of 2. No polytope of ... [more ▼] We determine, up to isomorphism and duality, the number of abstract regular polytopes of rank three, whose automorphism group is a Suzuki simple group Sz(q), with q an odd power of 2. No polytope of higher rank exists and therefore, the formula obtained counts all polytopes of Sz(q). Moreover, there are no degenerate polyhedra. We also obtain, up to isomorphism, the number of pairs of involutions. [less ▲] Detailed reference viewed: 11 (0 UL) |
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