Abelianization and fixed point properties of units in integral group rings

in Mathematische Nachrichten (2022)

#Discover Life on Mars with a Rover - Didaktischer Kommentar

E-print/Working paper (2022)

A dichotomy for integral group rings via higher modular groups as amalgamated products

in J. Algebra (2022), 604

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

in Groups, Geometry, and Dynamics (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 ▲]

Presentations of groups acting discontinuously on direct products of hyperbolic spaces

in Mathematics of Computation (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 ▲]

Describing units of integral group rings up to commensurability

in Journal of Pure and Applied 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 ▲]

On pairs of commuting involutions in $ Sym(n)$ and $ Alt(n)$

in Communications in 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 ▲]

On the number of abstract regular polytopes whose automorphism group is a Suzuki simple group $ Sz(q)$

in Journal of Combinatorial Theory. Series 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 ▲]