Thomas W. Müller

Decomposable Functors and the Exponential Principle

We develop a new setting for the exponential principle in the context of multisort species, where indecomposable objects are generated intrinsically instead of being given in advance. Our approach uses the language of functors and natural transformations (composition operators), and we show that, somewhat surprisingly, a single axiom for the composition already suffices to guarantee validity o the ex- ponential formula. We provide various illustrations of our theory, among which are applications to the enumeration of (semi-)magic squares.

Finite Group Actions and Asymptotic Expansion of e P(z) .

We establish an asymptotic expansion for the number |Hom (G,Sn)| of actions of a finite groupG on ann-set in terms of the order |G|=m and the numbersG(d) of subgroups of indexd inG ford|m. This expansion and related results on the enumeration of finite group actions follow from more general results concerning the asymptotic behaviour of the coefficients of entire functions of finite genus with finitely many zeros. As another application of these analytic considerations we establish an asymptotic property of the Hermite polynomials, leading to the explicit determination of the coefficientsCν(α;z) in Perrons asymptotic expansion for Laguerre polynomials in the cases α=±1/2.

Embedding theorems for tree-free groups

We establish two embedding theorems for tree-free groups. The first result embeds a group G acting freely and without inversions on a � -tree X into a groupacting freely, without inversions, and transitively on a � -treein such a way that X embeds intoby means of a G-equivariant isometry. The second result embeds a group G acting freely and transitively on an R-tree X into RF(H ) for some suitable group H , again in such a way that X embeds G-equivariantly into the R-tree XH associated with RF(H ). The group RF(H ) referred to here belongs to a class of groups introduced and studied by the present authors in (3). As a consequence of these two theorems, we find that RF-groups and their associated R-trees are in fact universal for free R-tree actions. Moreover, our first embedding theorem throws light on the question, arising from the results of (7), whether a group endowed with a Lyndon length function L can always be embedded in a length-preserving way into a group with a regular Lyndon length function; modulo an obvious necessary restriction we show that this is indeed the case if L is free.

Cyclic Sieving for Generalised Non-crossing Partitions Associated with Complex Reflection Groups of Exceptional Type

We prove that the generalised non-crossing partitions associated with well-generated complex reflection groups of exceptional type obey two different cyclic sieving phenomena, as conjectured by Armstrong, and by Bessis and Reiner. The computational details are provided in the manuscript “Cyclic sieving for generalised non-crossing partitions associated with complex reflection groups of exceptional type—the details” [arχiv:1001.0030].

A Group-theoretical Generalization of Pascal's Triangle

We introduce a class of infinite triangles with rational entries generalizing Pascals triangle. This construction is motivated by a recent investigation concerning the growth behaviour and the asymptotics of the number of torsion-free subgroups of finite index in a finitely generated virtually free group. We show that ‘most’ of these triangles enjoy the unimodal property, and we discern which triangles have the property that the sum of the reciprocal values of the entries in the interior of a row converges to 0.

Modular arithmetic of free subgroups

Abstract Denote by ƒλ (G ) the number of free subgroups of index λmG , where mG is the least common multiple of the orders of the finite subgroups in G. The present paper develops a general theory for the p-divisibility of ƒλ (G ), where p is a prime dividing mG . Among other things, we obtain an explicit combinatorial description of ƒλ (G ) modulo p, leading to an optimal generalisation of Stothers’ explicit formula for the parity of ƒλ (PSL2 (ℤ)).

Groups : topological, combinatorial and arithmetic aspects

1. Reductive groups as metric spaces H. Abels 2. Finiteness properties of groups acting on twin buildings P. Abramenko 3. Higher finiteness properties of S-arithmetic groups in the function field case I H. Behr 4. Controlled topology and group actions R. Bieri and R. Geoghegan 5. Finiteness properties of soluble S-arithmetic groups - a survey K. U. Bux 6. Topology in permutation groups P. Cameron 7. Euler characteristics of discrete groups I. Chiswell 8. Intersection of Magnus subgroups of one-relator groups D. J. Collins 9. A minimality property of certain branch groups R. I. Grigorchuk and J. S. Wilson 10. Lattices with non-integral character H. Helling 11. Some applications of probability in group theory A. Mann 12. Parity patterns in Hecke groups and Fermat primes T. W. Muller 13. Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms V. Nekrashevych and S. Sidki 14. The mapping class group of the twice punctured torus J. R. Parker and C. Series 15. Kac-Moody groups: split and relative theories. Lattices B. Remy 16. On the finite images of infinite groups D. Segal 17. Pseudo-finite generalized triangle groups E. B. Vinberg and R. Kaplinsky.

Global constitutionalism in historical perspective: Towards refined tools for international constitutional histories

This article raises the question of how the evolution of constitutional structures in international history can be conceptualized and narrated. This question is nurtured by the divergent historical accounts found in the literature on global constitutionalism and the historical-comparative literature on international societies. Although both literatures have so far largely ignored each other, they share a number of common conceptual problems and should therefore engage in a dialogue. The article sketches out the contours of such a dialogue to reveal the insufficient sensitivity of both literatures to the historicity of constitutional structures as well as their inherent political dimension. This article proposes to conceptualize constitutional structures as a set of fundamental and prioritized principles and rules that serves as a framework for the self-ordering of relations between polities. It thereby differentiates between a meta-constitutional and a legal-constitutional dimension and proposes seven analytical tools for more nuanced and empirically sensitive constitutional histories of interpolity relations. Such a historical perspective helps to account for the conditions of emergence as well as to delineate the novelty of contemporary global constitutionalism.

Equations in finite semigroups: explicit enumeration and asymptotics of solution numbers

We study the number of solutions of the general semigroup equation in one variable, Xα = Xβ, as well as of the system of equations X2 = X, Y2 = Y, XY = YX in H ≀ Tn, the wreath product of an arbitrary finite group H with the full transformation semigroup Tn on n letters. For these solution numbers, we provide explicit exact formulae, as well as asymptotic estimates. Our results concerning the first mentioned problem generalize earlier results by Harris and Schoenfeld (J. Combin. Theory Ser. A 3 (1967) 122) on the number of idempotents in Tn, and a result of Dress and the second author (Adv. in Math. 129 (1997) 188). Among the asymptotic tools employed are Haymans method for the estimation of coefficients of analytic functions and the Poisson summation formula.

Die Englische Schule in den Internationalen Beziehungen

Dieses Kapitel stellt die Englische Schule in den Internationalen Beziehungen vor. Das Kernargument dieser breiten theoretischen Tradition ist, dass Staaten durch gemeinsame Werte, Normen und Institutionen zu einer internationalen Gesellschaft integriert sind. Das Kapitel fuhrt nach einem kurzen Uberblick uber die Theoriegeschichte in die drei wichtigsten Konzepte internationales System, internationale Gesellschaft und Weltgesellschaft ein und geht anschliesend auf die wesentlichen Debatten in der Englischen Schule sowie ihre Verankerung in und Beitrage zu den Internationalen Beziehungen ein.