The subgroup structure of a direct product of two Fuchsian groups is very complicated; for example, Baumslag and Roseblade [1] have shown that a direct product Fm1×Fm2 of finitely generated free groups contains continuously many distinct isomorphism classes of finitely generated subgroups. The situation is much simpler, however, if attention is restricted to finitely generated normal subgroups of Fm1×Fm2; then on general grounds one cannot expect to get more than a countable infinity, but, in fact, the situation is almost finite; we showed, in [4], that Fm1×Fm2 contains precisely 1+min{m1, m2} orbits of maximal normal subdirect products under the natural action of its automorphism group.

In this paper we study the situation which arises if the free groups Fmi are replaced by fundamental groups of closed surfaces. This question was previously considered by Nigel Carr in the final chapter of his (unpublished) thesis [2]. By appealing to the relative invariant theory of pairs of skew bilinear forms, Carr was able to show that in a direct product Σg1×Σg2 of orientable surface groups of genus [ges ]2 the number of orbits of maximal normal subdirect products is always infinite. Here we refine and extend Carr's approach to study the manner in which the finiteness result of [4] breaks down on passing to more general Fuchsian groups.

The problem of constructing non-diagonal solutions to systems of symmetric diagonal equations has attracted intense investigation for centuries (see [5, 6] for a history of such problems) and remains a topic of current interest (see, for example, [2–4]). In contrast, the problem of bounding the number of such non-diagonal solutions has commanded attention only comparatively recently, the first non-trivial estimates having been obtained around thirty years ago through the sieve methods applied by Hooley [10, 11] and Greaves [7] in their investigations concerning sums of two kth powers. As a further contribution to the problem of establishing the paucity of non-diagonal solutions in certain systems of diagonal diophantine equations, in this paper we bound the number of non-diagonal solutions of a system of simultaneous quadratic and biquadratic equations. Let S(P) denote the number of solutions of the simultaneous diophantine equations

formula here

with 0[les ]xi, yi[les ]P(1[les ]i[les ]3), and let T(P) denote the corresponding number of solutions with (x1, x2, x3) a permutation of (y1, y2, y3). In Section 4 below we establish the upper and lower bounds for S(P)−T(P) contained in the following theorem.

Let [gscr ] be one of the Lie algebras [sscr ][Iscr ]n(K) or [sscr ][oscr ]2n+1(K) over an algebraically closed field K of characteristic p>0. Suppose in the first case that n∉Zp and in the second case that p≠2. This assumption implies that [gscr ] is simple. In this paper I study certain irreducible representations of [gscr ].

In this paper we consider reduced curves Γ of fixed degree d in the complex projective plane P2([Copf ]): we study the question of how singular such a curve can be.

This paper relates skein spaces based on the Kauffman bracket and spin structures. A spin structure on an oriented 3-manifold provides an isomorphism between the skein space for parameter A and the skein space for parameter −A.

There is an application to Penrose's binor calculus, which is related to the tensor calculus of representations of SU(2). The perspective developed here is that this tensor calculus is actually a calculus of spinors on the plane and the matrices are determined by a type of spinor transport which generalizes to links in any 3-manifold.

A second application shows that there is a skein space which is the algebra of functions on the set of spin structures for the 3-manifold.

In the early seventies, Steenrod posed the question: which polynomial algebras over the Steenrod algebra appear as the cohomology ring of a topological space? For odd primes, work of Adams and Wilkerson and Dwyer, Miller and Wilkerson showed that all such algebras are given as the mod-p reduction of the invariants of a pseudo reflection group acting on a polynomial algebra over the p-adic integers. We show that this necessary condition is also sufficient for finding a realization of such a polynomial algebra. We also show that, up to completion, every space with polynomial mod-p cohomology is equivalent to the product of the classifying space of a connected compact Lie group and some spaces determined by irreducible p-adic rational pseudo reflection groups.

In this paper we define a set of radii called thickness for simple closed curves denoted by K, which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these curves. One can think of these radii as representing the thickness of a rope in space and of K as the core of the rope. Great care is taken to define our radii in order to gain freedom from small pieces with large curvature in the curve. Intuitively, this means that we tend to allow the surface of the ropes that represent the knots to deform into a non smooth surface. But as long as the radius of the rope is less than the thickness so defined, the surface of the rope will remain a two manifold and the rope (as a solid torus) can be deformed onto K via strong deformation retract. In this paper we explore basic properties of these thicknesses and discuss the relationship amongst them.

In this paper we construct a symmetric monoidal smash product of Γ-spaces modelling the smash product of connective spectra. For the corresponding theory of ring-spectra, we refer the reader to [Sch].

In this paper we advertise the category of Γ-spaces as a convenient framework for doing ‘algebra’ over ‘rings’ in stable homotopy theory. Γ-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (−1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for Γ-spaces. The study of ‘rings, modules and algebras’ based on Γ-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg–MacLane spectra counterparts.

For a class of (n+1)-dimensional manifolds of hyperbolic type, the spectral measure is described in terms of Poisson kernels. This implies that the spectrum is absolutely continuous and it is contained in [n2/4; +∞).

We study the action of fractional differentiation and integration on weighted Bergman spaces and also the Taylor coeffficients of functions in certain subclasses of these spaces. We then derive several criteria for the multipliers between such spaces, complementing and extending various recent results. Univalent Bergman functions are also considered.