A cohomological property of <Emphasis Type="Italic">p</Emphasis>-groups

Let p be a prime, a finite p-group, any finite group with order divisible by p, and any action of on . We show that the cardinality of the set of all derivations with respect to this action is a multiple of p. This generalises theorems of Frobenius and Hall. Received: 16 June 2003     

Reduced Norm Map of Division Algebras over Complete Discrete Valuation Fields of Certain Type

We study a ramification theory for a division algebra D of the following type: The center of D is a complete discrete valuation field K with the imperfect residue field F of certain type, and the residue algebra of D is commutative and purely inseparable over F. Using Swan conductors of the corresponding element of Br(K), we define -function of D/K, and it describe the action of the reduced norm map on the filtration of D-. We also gives a relation among the Swan conductors and the ramification number of D, which is defined by the commutator group of D-.     

Holomorphic Torus Actions on Compact Locally Conformal Kähler Manifolds

Given a torus action (T ², M) on a smooth manifold, the orbit map ev _x(t)=t·xfor each xMinduces a homomorphism ev _*: ²H ₁(M;). The action is said to be Rank-kif the image of ev _*has rank k(2) for each point of M. In particular, if ev _*is a monomorphism, then the action is called homologically injective. It is known that a holomorphic complex torus action on a compact Kähler manifold is homologically injective. We study holomorphic complex torus actions on compact non-Kähler Hermitian manifolds. A Hermitian manifold is said to be a locally conformal Kähler if a lift of the metric to the universal covering space is conformal to a Kähler metric. We shall prove that a holomorphic conformal complex torus action on a compact locally conformal Kähler manifold Mis Rank-1 provided that Mhas no Kähler structure.     

The Hodge Structure on a Filtered Boolean Algebra

Let (B _n) be the order complex of the Boolean algebra and let B(n, k) be the part of (B _n) where all chains have a gap at most k between each set. We give an action of the symmetric group S _l on the l-chains that gives B(n, k) a Hodge structure and decomposes the homology under the action of the Eulerian idempontents. The S _n action on the chains induces an action on the Hodge pieces and we derive a generating function for the cycle indicator of the Hodge pieces. The Euler characteristic is given as a corollary.We then exploit the connection between chains and tabloids to give various special cases of the homology. Also an upper bound is obtained using spectral sequence methods.Finally we present some data on the homology of B(n, k).     

Actions of SL $$(n,\mathbb{Z})$$ on Homology Spheres

Any continuous action of SL , where n > 2, on a r-dimensional mod 2 homology sphere factors through a finite group action if r < n −1. In particular, any continuous action of SL on the n-dimensional sphere factors through a finite group action.Mathematics Subject Classiffications (2000). Primary 57S25; Secondary 37C85, 57S17     

Topological Properties of SL 2 Actions on the Hyperbolic Plane

We describe new topological invariants of the classical action of SL ₂([1/m]) on the hyperbolic plane, where m is a positive integer. One defines the notion of this action (or any action of a discrete group on a CAT(0) metric space) as being 'controlled s-connected' where s is an integer –1. In the case of this classical action, we show it is (s–2)-connected but not (s–1)-connected where s is the number of different primes dividing m. It turns out that the behavior of this action is quite different over rational and irrational endpoints, with the bounds on connectivity occurring only at the rational end points. The very lowest case, m=1, reduces to elementary and seemingly innocuous statements about how SL ₂() acts on the hyperbolic plane. Extensions involving other Fuchsian groups are also given.     

Final Lift Actions Associated with Topological Functors

For a topological functor U:EB, the fiber U ^–1(b), bB, is a cocomplete poset and the left action, induced by final lift, of the endomorphism monoid B(b,b) on U ^–1(b) is cocontinuous. It is shown that every cocontinuous left action of B(b,b) on any cocomplete poset can be realized as the final lift action associated to a canonically defined topological functor over B. If B is a Grothendieck topos and b=, the subobject classifier, then B(,) inherits both a monoidal and a cocomplete poset structure. In the case B= Sets, all cocontinuous left actions of B(,) on itself are explicitly described and each is shown to arise as the final lift action associated to a specific subcategory of a certain fixed category, referred to as the category of LR-spaces. Relationships between these LR-spaces and several other well known topological categories are also considered.     

A Hochschild Homology Euler Characteristic for Circle Actions

We define an S¹-Euler characteristic, _S ¹(X), of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group HH ₁(G) where G is the fundamental group of X. This _S ¹(X) is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for the Euler class (integer coefficients) of the normal bundle to a smooth circle action without fixed points on a manifold. In the special case of a three-dimensional Seifert fibered space, this formula is particularly effective.     

Finite splitting fields of normal subgroups

Let G be a finite group, a normal subgroup, p a prime, a finite splitting field of characteristic p for G and We prove that is a splitting field for N, using the action of the Galois group of the field extension on the irreducible representations of N. As is a splitting field for the symmetric group S_n we get as a corollary that is a splitting field for the alternating group A_n. Received: 31 July 2003     

Approximations of stable actions on $ \Bbb {R} $-trees

This article shows how to approximate a stable action of a finitely presented group on an -tree by a simplicial one while keeping control over arc stabilizers. For instance, every small action of a hyperbolic group on an -tree can be approximated by a small action of the same group on a simplicial tree. The techniques we use highly rely on Rips's study of stable actions on -trees and on the dynamical study of exotic components by D. Gaboriau. Received: 22 October, 1997     

Improper actions and higher connectivity at infinity

Given an improper action (= cell stabilizers are infinite) of a group G on a CW-complex , we present criteria, based on connectivity at infinity properties of the cell stabilizers under the action of G that imply connectivity at infinity properties for G. A refinement of this idea yields information on the topology at infinity of Artin groups, and it gives significant progress on the question of which Artin groups are duality groups. Received: October 30, 1998     

Cocyclic Development of Designs

We present the basic theory of cocyclic development of designs, in which group development over a finite group G is modified by the action of a cocycle defined on G × G. Negacyclic and -cyclic development are both special cases of cocyclic development.Techniques of design construction using the group ring, arising from difference set methods, also apply to cocyclic designs. Important classes of Hadamard matrices and generalized weighing matrices are cocyclic.We derive a characterization of cocyclic development which allows us to generate all matrices which are cocyclic over G. Any cocyclic matrix is equivalent to one obtained by entrywise action of an asymmetric matrix and a symmetric matrix on a G-developed matrix. The symmetric matrix is a Kronecker product of back -cyclic matrices, and the asymmetric matrix is determined by the second integral homology group of G. We believe this link between combinatorial design theory and low-dimensional group cohomology leads to (i) a new way to generate combinatorial designs; (ii) a better understanding of the structure of some known designs; and (iii) a better understanding of known construction techniques.     

Equivariant Reduction to Torus of a Principal Bundle

Let M be an irreducible projective variety, over an algebraically closed field k of characteristic zero, equipped with an action of a connected algebraic group S over k. Let E _G be a principal G-bundle over M equipped with a lift of the action of S on M, where G is a connected reductive linear algebraic group. Assume that E _G admits a reduction of structure group to a maximal torus TG. We give a necessary and sufficient condition for the existence of a T-reduction of E _G which is left invariant by the action of S on E _G .     

Intersection queries in sets of disks

In this paper we develop some new data structures for storing a set of disks that can answer different types of intersection queries efficiency. If the disks are non-intersecting we obtain a linear size data structure that can report allk disks intersecting a query line segment in timeO(n ⁺ +k), wheren is the number of disks,=log₂(1+5)–1 0.695, and is an arbitrarily small positive constant. If the segment is a full line, the query time becomesO(n +k). For intersecting disks we obtain anO(n logn) size data structure that can answer an intersection query in timeO(n ^2/3 log² n+k). We also present a linear size data structure for ray shooting queries, whose query time isO(n ).The research of the first two authors was supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM). The work of the third author was supported byDimacs (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center — NSF-STC88-09648.     

Editorial Addresses

We study the existence and structure of extremals for one-dimensional variational problems on a torus and the properties of the minimal average action as a function of the rotation number. We show that, for a generic integrand f, the minimum of the minimal average action is attained at a rational point mn ^–1 where n1 and m are integers; also, for each initial value, there exists an (f)-weakly optimal solution over an infinite horizon.     

A geometric construction of local representations of local Lie groups

Local transformation groups acting on a manifold X define a natural action of on a space D(X), of functions on X. The natural action induces a local representation of on a Hilbert subspace of the space of distributions on D(X).     

The Drinfel’d Double for Group-cograded Multiplier Hopf Algebras

Let G be any group and let K(G) denote the multiplier Hopf algebra of complex functions with finite support in G. The product in K(G) is pointwise. The comultiplication on K(G) is defined with values in the multiplier algebra M(K(G) ⊗K(G )) by the formula for all and . In this paper we consider multiplier Hopf algebras B (over ) such that there is an embedding I: K(G) →M(B). This embedding is a non-degenerate algebra homomorphism which respects the comultiplication and maps K(G) into the center of M(B). These multiplier Hopf algebras are called G-cograded multiplier Hopf algebras. They are a generalization of the Hopf group-coalgebras as studied by Turaev and Virelizier. In this paper, we also consider an admissible action π of the group G on a G-cograded multiplier Hopf algebra B. When B is paired with a multiplier Hopf algebra A, we construct the Drinfel’d double D ^π where the coproduct and the product depend on the action π. We also treat the ^*-algebra case. If π is the trivial action, we recover the usual Drinfel’d double associated with the pair . On the other hand, also the Drinfel’d double, as constructed by Zunino for a finite-type Hopf group-coalgebra, is an example of the construction above. In this case, the action is non-trivial but related with the adjoint action of the group on itself. Now, the double is again a G-cograded multiplier Hopf algebra. Presented by K. Goodearl.     

Invariants of 2 × 2 matrices, irreducible {\hbox{SL}(2, \mathbb{C})} characters and the Magnus trace map

We obtain an explicit characterization of the stable points of the action of on the cartesian product G  ^{× n} by simultaneous conjugation on each factor in terms of the corresponding invariant functions. From this, a simple criterion for the irreducibility of representations of finitely generated groups into G is derived. We also obtain analogous results for the action of on the vector space of n-tuples of 2 × 2 complex matrices. For a free group F _n of rank n, we show how to generically reconstruct the 2 ^n-2 conjugacy classes of representations F _n → G from their values under the map considered in Magnus [Math. Zeit. 170, 91–103 (1980)], defined by certain 3n − 3 traces of words of length one and two.      

Parametrization of the orbits of cubic surfaces

The aim of the paper is the study of the orbits of the action of PGL₄ on the space ³ of the cubic surfaces of ³, i.e., the classification of cubic surfaces up to projective motions. A varietyQ¹⁹ is explicitely constructed as the union of 22 disjoint irreducible components which are either points or open subsets of linear spaces. More precisely, each orbit of the above action intersects one componentX ofQ in a finite number of points and the action of PGL₄ restricted on each componentX is equivalent to the action of a finite groupG _X onX which can be explicitely computed. Finally the cubic surfaces of each component ofQ are studied in details by determining their stabilizers, their rational representations and whether they can be expressed as the determinant of a 3×3 matrix of linear forms.The results are obtained with computational techniques and with the aid of some computer algebra systems like CoCoA, Macaulay and Maple.Partially supported by MURSTPartially supported by MURST and CNR