Volume 10, Issue 6

In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and torsion free. We connect the Reidemeister zeta function of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group with the Lefschetz zeta function of the induced map on the unitary dual of the group. As a consequence we obtain a relation between a special value of the Reidemeister zeta function and a certain Reidemeister torsion. We also prove congruences for Reidemeister numbers of iterates of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group which are the same as those found by Dold for Lefschetz numbers.

In this article we first prove a stability theorem for coverings in 𝔼 2 by congruent solid circles: if the density of such a covering is close to its lower bound , then most of the centers of the circles are arranged in almost regular hexagonal patterns. A version of this result then is extended to coverings by geodesic discs in two-dimensional Riemannian manifolds. Given a sufficiently differentiable convex body C in 𝔼 3 , the following two problems are closely related: (i) Approximation of C with respect to the Hausdorff metric, the Banach-Mazur distance and a notion of distance due to Schneider by inscribed or circumscribed convex polytopes. (ii) Covering of the boundary of C by geodesic discs with respect to suitable Riemannian metrics. The stability result for Riemannian manifolds and the relation between approximation and covering yield rather precise information on the form of best approximating inscribed convex polytopes P n of C with respect to the Hausdorff metric: if the number n of vertices is large, then most of the vertices are arranged in almost regular hexagonal patterns. Consequently, the majority of facets of P n are almost regular triangles. Here ‘regular’ is meant with respect to the Riemannian metric of the second fundamental form. Similar results hold for circumscribed polytopes and also for the Banach-Mazur distance and Schneider's notion of distance.

Given a probability distribution μ a set Λ ( μ ) of positive real numbers is introduced, so that Λ ( μ ) measures the “divisibility” of μ . The basic properties of Λ ( μ ) are described and examples of probability distributions are given, which exhibit the existence of a continuum of situations interpolating the extreme cases of infinitely and minimally divisible probability distributions.

For a convex body K in Euclidean space ℝ d ( d ≥2) and for r ∈ {0,…, d – 1}, let Σ r ( K ) be the set of r -singular boundary points of K . It is known that Σ r ( K ) is countably r -rectifiable and hence has σ-finite r -dimensional Hausdorff measure. We obtain a quantitative improvement of this result, taking into account the strength of the singularities. Denoting by Σ r ( K , τ) the set of those r -singular boundary points of K at which the spherical image has ( d –1– r )-dimensional Hausdorff measure at least τ>0, we establish a finite upper bound for the r -dimensional Hausdorff measure of Σ r ( K , τ). This estimate is deduced from an identity that connects Hausdorff measures of spherical images of singularities to the generalized curvature measure of the convex body K . The latter relation is, in fact, proved for the class of sets with positive reach. For convex bodies, similar result as for singular boundary points are obtained for singular normal vectors. We also consider the disintegration of generalized curvature measures with respect to projections onto the components of the product space ℝ d × S d – 1 .

An almost completely decomposable (acd) group is afiniterank torsion-free abelian group containing a finite direct sum of rank-1 groups as a subgroup of finite index. This paper is devoted to a determination of the representation type of indecomposables in categories of acd groups. In the process, representation types of categories of related representations of finite partially ordered sets (posets) over the rings Z/p j Z and Z p , the localization of the integers Z at a prime p , are also determined.

Smooth stable planes have been introduced in [4]. We show that every continuous collineation between two smooth stable planes is in fact a smooth collineation. This implies that the group Γ of all continuous collineations of a smooth stable plane is a Lie transformation group on both the set P of points and the set ℒ of lines. In particular, this shows that the point and line sets of a (topological) stable plane ℐ admit at most one smooth structure such that ℐ becomes a smooth stable plane. The investigation of central and axial collineations in the case of (topological) stable planes due to R. Löwen ([25], [26], [27]) is continued for smooth stable planes. Many results of [26] which are only proved for low dimensional planes (dim ℐ ≤ 4) are transferred to smooth stable planes of arbitrary finite dimension. As an application of these transfers we show that the stabilizers Γ [ c,c ] 1 and Γ [ A,A ] 1 (see (3.2) Notation) are closed, simply connected, solvable subgroups of Aut(ℐ) (Corollary (4.17)). Moreover, we show that Γ [ c,c ] is even abelian (Theorem (4.18)). In the closing section we investigate the behaviour of reflections.