Let G be a group endowed with its profinite topology, then G is called product separable if the profinite topology of G is Hausdorff and, whenever H1, H2, [ctdot ], Hn are finitely generated subgroups of G, then the product subset H1H2 [ctdot ] Hn is closed in G. In this paper, we prove that if G=F×Z is the direct product of a free group and an infinite cyclic group, then G is product separable. As a consequence, we obtain the result that if G is a generalized free product of two cyclic groups amalgamating a common subgroup, then G is also product separable. These results generalize the theorems of M. Hall Jr. (who proved the conclusion in the case of n=1, [3]), and L. Ribes and P. Zalesskii (who proved the conclusion in the case of that G is a finite extension of a free group, [6]).

Let G be a permutation group on a set Ω, and let m and k be integers where 0<m<k. For a subset Γ of Ω, if the cardinalities of the sets Γg\Γ, for g∈G, are finite and bounded, then Γ is said to have bounded movement, and the movement of Γ is defined as move (Γ) =maxg∈G[mid ]Γg\Γ[mid ]. If there is a k-element subset Γ such that move (Γ)[les ]m, it is shown that some G-orbit has length at most (k2−m)\(k−m). When combined with a result of P. M. Neumann, this result has the following consequence: if some infinite subset Γ has bounded movement at most m, then either Γ is a G-invariant subset with at most m points added or removed, or Γ nontrivially meets a G-orbit of length at most m2+m+1. Also, if move (Γ)[les ]m for all k-element subsets Γ and if G has no fixed points in Ω, then either [mid ]Ω[mid ][les ]k+m (and in this case all permutation groups on Ω have this property), or [mid ]Ω[mid ][les ]5m−2. These results generalise earlier results about the separation of finite sets under group actions by B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann, and groups in which all subsets have bounded movement (by the author).

In this paper we are concerned with abstract homomorphisms from one algebraic group to another. That is, group homomorphisms where there is no assumption of rationality. Let K be an infinite perfect field and L an algebraically closed field of the same characteristic. Throughout G(K) will denote a Chevalley group of universal type over K.

We prove a complex analytic analogue of the classical Rolle theorem asserting that the number of zeros of a real smooth function can exceed that of its derivative by at most 1. This result is used then to obtain upper bounds for the number of complex isolated zeros of:

(1) functions defined by linear ordinary differential equations (in terms of the magnitude of the coefficients of the equations);

(2) elements from the polynomial envelope of a linear differential equation with an irreducible monodromy group (in terms of the degree of the envelope);

(3) successive derivatives of a function defined by a linear irreducible equation (in terms of the order of the derivative).

These results generalize the bounds from [2, 5, 6] that were previously obtained for the number of real isolated zeros.

We give a complete description of the topological spaces X such that the free abelian topological group A(X) embeds into the free abelian topological group A(I) on the closed unit interval. In particular, the free abelian topological group A(X) on any finite-dimensional compact metrizable space X embeds into A(I). To obtain our description, we study similar embeddings of the free locally convex spaces and continuous surjections between the spaces of continuous functions with the pointwise topology. Proofs are based on the classical Kolmogorov's Superposition Theorem.

In this paper we consider the form of the eigenvalue counting function ρ for Laplacians on p.c.f. self-similar sets, a class of self-similar fractal spaces. It is known that on a p.c.f. self-similar set K the function ρ(x)=O(xds/2) as x→∞, for some ds>0. We show that if there exist localized eigenfunctions (that is, a non-zero eigenfunction which vanishes on some open subset of the space) and K satisfies some additional conditions (‘the lattice case’) then ρ(x)x−ds/2 does not converge as x→∞. We next establish a number of sufficient conditions for the existence of a localized eigenfunction in terms of the symmetries of the space K. In particular, we show that any nested fractal with more than two essential fixed points has localized eigenfunctions.

A group is accessible if there is a G-tree T such that every edge stabilizer is finite and every vertex stabilizer has at most one end. A group is inaccessible if it is not accessible. In [6] I began the study of group actions on protrees. These actions arise naturally when considering inaccessible groups. This study is continued in this paper.

In 1940 Nisnevič published the following theorem [3]. Let (Gα)α∈Λ be a family of groups indexed by some set Λ and (Fα)α∈Λ a family of fields of the same characteristic p[ges ]0. If for each α the group Gα has a faithful representation of degree n over Fα then the free product *α∈ΛGα has a faithful representation of degree n+1 over some field of characteristic p. In [6] Wehrfritz extended this idea. If (Gα)α∈Λ [les ]GL(n, F) is a family of subgroups for which there exists Z[les ]GL(n, F) such that for all α the intersection Gα∩F.1n=Z, then the free product of the groups *ZGα with Z amalgamated via the identity map is isomorphic to a linear group of degree n over some purely transcendental extension of F.

Initially, the purpose of this paper was to generalize these results from the linear to the skew-linear case, that is, to groups isomorphic to subgroups of GL(n, Dα) where the Dα are division rings. In fact, many of the results can be generalized to rings which, although not necessarily commutative, contain no zero-divisors. We have the following.

We consider operators on the matrix-valued disc algebra. We show that any bounded linear operator T: A[otimes ][Kscr ]→X to a Banach space X of cotype 2 induces a bounded operator Gν→X defined on some weighted Bergman space Gν on the unit disc. We give sufficient conditions on the weight ν for the formal inclusion A[otimes ][Kscr ]→Gν to be 2−C*-summing. Decompositions of T with respect to operator-valued measures are obtained.

An integral of the generalized Riemann type is developed which inverts the Schwarz derivative of a continuous function.

Let G be a finite group and k be an algebraically closed field of prime characteristic. Corresponding to each closed homogeneous subvariety W of the maximal ideal spectrum of H*(G, k) we construct (usually infinite-dimensional) kG-modules E(W) and F(W) which are idempotent in the sense that E(W) and F(W) are isomorphic (up to projective summands) to E(W) [otimes ] E(W) and F(W) [otimes ] F(W) respectively. We study the properties of these modules, and as an application we use them to describe natural direct sum decompositions of modules in quotient categories.

In the last decade much research has been done on questions related to circle packings, which are collections of circles in the plane with prescribed patterns of tangencies encoded by simplicial 2-complexes. The interest in circle packings and their connections with analytic functions was initiated by W. Thurston in 1985 when he suggested a method for approximation of conformal mappings using circle packings [20]. This method was shown to work by B. Rodin & D. Sullivan [15] later that year (see also [17]). Since then several results have been published (for example, [3, 4, 5, 10, 11, 14]) which seem to indicate the possibility of creating a discrete analog of the theory of analytic functions via circle packings. However, most of these results deal with univalent circle packings, that is, genuine packings of circles. Influenced by the idea of [3] that circle packings might be used to construct discrete parallels of analytic functions, we started in [7] to investigate non-univalent circle packings, particularly branched packings. We introduced there the notion of discrete Blaschke products and proved a branched version of the finite Riemann mapping theorem (see [20, 15]). We feel though that to have a solid foundation for a circle packing analog of the theory of analytic functions one needs something more. Since complex polynomials form a fundamental and very important class of analytic functions, it is natural to try to establish circle packing counterparts of these mappings. This is exactly the main theme of our paper.

For every Hausdorff function h, the paper gives a construction of a compact metric space of infinite diameter-based h-packing measure which has no subsets of positive finite measure. It then indicates how such a construction can be modified to provide the same result for a radius-based packing measure in the case of certain Hausdorff functions which do not satisfy a doubling condition. This is in surprising contrast to another result which shows that for a particular radius-based definition of packing measure no such example can be found.

We render the cut locus more accessible to analysis by showing that each non-conjugate cut point has a neighbourhood on which the distance function is the minimum of finitely many smooth ‘radial functions’. This enables us to generalise to an arbitrary complete manifold a number of recent results, mainly from stochastic analysis, which were either limited or not valid on the cut locus because of the lack of differentiability there of the distance function.

Throughout ℤp and ℚp will, respectively, denote the ring of p-adic integers and the field of p-adic numbers (for p prime). We denote by [Copf ]p the completion of the algebraic closure of ℚp with respect to the p-adic metric. Let vp denote the p-adic valuation of [Copf ]p normalised so that vp(p)=1. Put [ ]p ={ω∈[Copf ]p[mid ] ωpn=1 for some n[ges ]0} so that [ ]p is the union of cyclic (multiplicative) groups Cpn of order pn (for n[ges ]0).

Let UD(ℤp) denote the [Copf ]p-algebra of all uniformly differentiable functions f[ratio ]ℤp→[Copf ]p under pointwise addition and convolution multiplication *, where for f, g∈UD(ℤp) and z∈ℤp we have

formula here

the summation being restricted to i, j with vp(i+j−z)[ges ]n.

This situation is a starting point for p-adic Fourier analysis on ℤp, the analogy with the classical (complex) theory being substantially complicated by the absence of a p-adic valued Haar measure on ℤp (see [5, 6] for further details).

A well-known theorem of Hurwitz states that if τ[ratio ]S→S is a conformal self-mapping of a compact Riemann surface of genus g[ges ]2, then it has at most 2g+2 fixed points and that equality occurs if and only if τ is a hyperelliptic involution.

In this paper we consider this problem for a K-quasiconformal self-mapping f[ratio ]S→S. The result we obtain is that the number of fixed points (suitably counted) is bounded by 2+g(K1/2+K−1/2), and that this bound is sharp. We see that when K=1, that is, when f is conformal, our result agrees with the classical one.

Let θ be an irrational number in [0, 1] and Aθ the corresponding irrational rotation C*-algebra. Let Aut (Aθ) be the group of all automorphisms of Aθ and Int (Aθ) the normal subgroup of Aut (Aθ) of all inner automorphisms of Aθ. Let Pic (Aθ) be the Picard group of Aθ. In the present note we shall show that if θ is not quadratic, then Pic (Aθ)≅Aut (Aθ)/Int (Aθ) and that if θ is quadratic, then Pic (Aθ) is isomorphic to a semidirect product of Aut (Aθ)/Int (Aθ) with ℤ. Furthermore, in the last section we shall discuss Picard groups of certain Cuntz algebras.

A construction is described of a 153-dimensional representation of the triple cover of O'Nan's sporadic simple group, over the field of order 4. This is then extended to a 306-dimensional representation of 3·O'N[ratio ]2 over GF(2). A similar construction in characteristic 3 gives an unrelated representation of the O'Nan group in 154 dimensions over GF(3), which extends to O'N[ratio ]2 provided that the field is extended to GF(9).

On sait associer à certaines structures de Poisson sur ℝn, de 1-jet nul en 0, des actions de ℝ2 sur ℝn, données par le ‘rotationnel’ de leur partie quadratique et un autre champ de vecteurs. Lorsque ces actions sont ‘non résonantes’ et ‘hyperboliques’, on montre que ces structures sont ‘quadratisables’, en ce sens qu'il existe des coordonnées dans lesquelles, elles sont quadratiques. Dans le cas de la dimension 3, nos résultats mènent à la ‘non-dégénérescence’ générique des structures de Poisson quadratiques à rotationnels inversibles.

We can associate with some Poisson structures defined on ℝn with a zero 1-jet at zero, actions from ℝ2 on ℝn, given by the ‘curl’ of their quadratic part and another vector field. Assuming that those actions are ‘hyperbolics’ and without ‘resonances’, we give a normal form for those structures. On ℝ3, we prove that every quadratic Poisson structure with invertible curl, is generically ‘non degenerate’.

We introduce the notion of the finite dimensional approximation property (the FDAP) and prove that if a subset X of a linear metric space has the FDAP, then every non-empty convex subset of X is an AR.

As an application we show that every needle point space X contains a dense linear subspace E with the following properties:

(i) E contains a non-empty compact convex set with no extreme points;

(ii) all non-empty convex subsets of E are AR.