On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

On the Hodge–Newton filtration for p-divisible groups of Hodge type

Mathematische Zeitschrift - A p-divisible group, or more generally an F-crystal, is said to be Hodge–Newton reducible if its Newton polygon and Hodge polygon have a nontrivial contact point....     

Classification of four-dimensional real Lie bialgebras of symplectic type and their Poisson–Lie groups

We classify all four-dimensional real Lie bialgebras of symplectic type and obtain the classical r-matrices for these Lie bialgebras and Poisson structures on all the associated four-dimensional Poisson–Lie groups. We obtain some new integrable models where a Poisson–Lie group plays the role of the phase space and its dual Lie group plays the role of the symmetry group of the system.     

Orderings of Witzel–Zaremsky–Thompson groups

We prove the orderability of the Witzel-Zaremsky-Thompson group for a direct system of orderable groups under a certain compatibility assumption.     

New “large” groups of type I

All unitary representations of certain infinite-dimensional, completely disconnected groups of type I are described.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 16, pp. 31–52, 1980.     

Rigidity of quantum tori and the Andruskiewitsch–Dumas conjecture

We prove the Andruskiewitsch–Dumas conjecture that the automorphism group of the positive part of the quantized universal enveloping algebra ${\mathcal {U}}_q({\mathfrak {g}})$ of an arbitrary finite dimensional simple Lie algebra ${\mathfrak {g}}$ is isomorphic to the semidirect product of the automorphism group of the Dynkin diagram of ${\mathfrak {g}}$ and a torus of rank equal to the rank of ${\mathfrak {g}}$ . The key step in our proof is a rigidity theorem for quantum tori. It has a broad range of applications. It allows one to control the (full) automorphism groups of large classes of associative algebras, for instance quantum cluster algebras.     

Assouad–Nagata dimension of connected Lie groups

We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.     

Rigidity of the Álvarez class

Let ${(M,\mathcal{F})}$ be a closed manifold with a Riemannian foliation. The Álvarez class of ${(M,\mathcal{F})}$ is a cohomology class of M of degree 1 whose triviality characterizes the minimizability or the geometrically tautness of ${(M,\mathcal{F})}$ . We show that the integral of the Álvarez class of ${(M,\mathcal{F})}$ along every closed path is the logarithm of an algebraic integer if π₁ M is polycyclic or ${\mathcal{F}}$ is of polynomial growth.     

Quasi-antichain Chermak–Delgado lattices of finite groups

The Chermak–Delgado lattice of a finite group is a dual, modular sublattice of the subgroup lattice of the group. This paper considers groups with a quasi-antichain interval in the Chermak–Delgado lattice, ultimately proving that if there is a quasi-antichain interval between subgroups L and H with L ≤ H then there exists a prime p such that H/L is an elementary abelian p-group and the number of atoms in the quasi-antichain is one more than a power of p. In the case where the Chermak–Delgado lattice of the entire group is a quasi-antichain, the relationship between the number of abelian atoms and the prime p is examined; additionally, several examples of groups with a quasi-antichain Chermak–Delgado lattice are constructed.     

On loxodromic actions of Artin–Tits groups

Artin–Tits groups act on a certain delta-hyperbolic complex, called the “additional length complex”. For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mapping class groups, we conjecture that “most” elements of Artin–Tits groups act loxodromically. More precisely, in the Cayley graph of a subgroup G of an Artin–Tits group, the proportion of loxodromically acting elements in a ball of large radius should tend to one as the radius tends to infinity. In this paper, we give a condition guaranteeing that this proportion stays away from zero. This condition is satisfied e.g. for Artin–Tits groups of spherical type, their pure subgroups and some of their commutator subgroups.     

On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C $ , let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$ . In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F $ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$ . We illustrate our results with the example of rank two Kac–Moody groups.     

Sharp weighted Hardy type inequalities and Hardy–Sobolev type inequalities on polarizable Carnot groups

In this Note, we establish sharp weighted Hardy type inequalities with a more general index p on polarizable Carnot groups, which include Kombe's recent results; then a weighted Hardy–Sobolev type inequality is obtained by using previous inequalities. To cite this article: J. Wang, P. Niu, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Necessary Conditions for the Global Rigidity of Direction–Length Frameworks

It is an intriguing open problem to give a combinatorial characterisation or polynomial algorithm for determining when a graph is globally rigid in ℝ ^d . This means that any generic realisation is uniquely determined up to congruence when each edge represents a fixed length constraint. Hendrickson gave two natural necessary conditions, one involving connectivity and the other redundant rigidity. In general, these are not sufficient, but they do suffice in two dimensions, as shown by Jackson and Jordán. Our main result is an analogue of the redundant rigidity condition for frameworks that have both direction and length constraints. For any generic globally rigid direction-length framework in ℝ ^d with at least 2 length edges, we show that deleting any length edge results in a rigid framework. It seems harder to obtain a corresponding result when a direction edge is deleted: we can do this in two dimensions, under an additional hypothesis that we believe to be unnecessary. Our proofs use a lemma of independent interest, stating that a certain space parameterising equivalent frameworks is a smooth manifold. We prove this lemma using arguments from differential topology and the Tarski–Seidenberg theorem on semi-algebraic sets.     

Biplanes with flag-transitive automorphism groups of almost simple type,with exceptional socle of Lie type

In this paper we prove that there is no biplane admitting a flag-transitive automorphism group of almost simple type, with exceptional socle of Lie type. A biplane is a (v,k,2)-symmetric design, and a flag is an incident point-block pair. A group G is almost simple with socle X if X is the product of all the minimal normal subgroups of G, and X⊴G≤Aut (G). Throughout this work we use the classification of finite simple groups, as well as results from P.B. Kleidman’s Ph.D. thesis which have not been published elsewhere.     

Metric properties of classes of Hölder surfaces on Carnot groups

Area formulas for classes of Hölder mappings of Carnot groups and the corresponding graph mappings are obtained. The calculation of a nonintrinsic measure is exemplified.