Ann Kiefer

• Vrije Universiteit Brussel

The Gruenberg–Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices p, q are joined by an edge whenever the group has an element of order pq. It reflects interesting properties of the group. A group is said to be cut if the central units of its integ...

Let G be a finite group and U(ZG)${\mathcal {U}}({\mathbb {Z}}G)$ the unit group of the integral group ring ZG${\mathbb {Z}}G$. We prove a unit theorem, namely, a characterization of when U(ZG)$\mathcal {U}(\mathbb {Z}G)$ satisfies Kazhdan's property (T)$(\operatorname{T})$, both in terms of the finite group G and in terms of the simple components...

We show that U(ZG), the unit group of the integral group ring ZG, either satisfies Kazhdan's property (T) or is, up to commensurability, a non-trivial amalgamated product, in case G is a finite group satisfying some mild conditions. A key step in the proof is the construction of amalgamated decompositions of the elementary group E2(O), where O is a...

The Gruenberg-Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices $p$, $q$ are joined by an edge whenever the group has an element of order $pq$. It reflects interesting properties of the group. A group is said to be cut if the central units of its...

The fixed point properties and abelianization of arithmetic subgroups $\Gamma$ of $\operatorname{SL}_n(D)$ and its elementary subgroup $\operatorname{E}_n(D)$ are well understood except in the degenerate case of lower rank, i.e. $n=2$ and $\Gamma = \operatorname{SL}_2(\mathcal{O})$ with $\mathcal{O}$ an order in a division algebra $D$ with only a f...

We give a concrete presentation for the general linear group defined over a ring which is a finitely generated free $\mathbb{Z}$-module or the integral Clifford group $\Gamma_n(\mathbb{Z})$ of invertible elements in the Clifford algebra with integral coefficients. We then use this presentation to prove that the elementary linear group over $\Gamma_...

We generalize an algorithm established in earlier work 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 inde...

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 $\...

We give a new self-contained proof of Poincar\'e's Polyhedron Theorem on presentations of discontinuous 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...

We continue the investigations of Lakeland on Fuchsian and Kleinian groups which have a Dirichlet fundamental domain that also is a Ford domain in hyperbolic two- or three-space; such a domain is called a DF-domain. Making use of earlier obtained concrete formulas for the bisectors in the upper half-space model, we obtain an easy algebraic criteriu...

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 isomorphism and duality, there are exactly two abstract regular polyhedra on which the group Sym(6) acts as a regular automorphism group.

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\mathbb{Z} G$ of a finite nilpotent group $G$, this provided the rational group algebra $\mathbb{Q} G$ does not have simple components that are division classical quaternion algebras or two-by-two matrices over a cl...

We determine explicit formulas for the bisectors used in constructing a Dirichlet fundamental domain in hyperbolic two and three space. They are compared with the isometric spheres employed in the construction of a Ford domain and used to find a finite set of generators for discrete groups of finite covolume. Applications are given to Fuchsian grou...

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 abstract regular polytopes of Sz(q). Moreover, there are no degenerate polyhedra. We...