It is known from Vaughan and Wooley's work on Waring's problem that every sufficiently large natural number is the sum of at most 17 fifth powers [13]. It is also known that at least six fifth powers are required to be able to express every sufficiently large natural number as a sum of fifth powers (see, for instance, [5, Theorem 394]). The techniques of [13] allow one to show that almost all natural numbers are the sum of nine fifth powers. A problem of related interest is to obtain an upper bound for the number of representations of a number as a sum of a fixed number of powers. Let R(n) denote the number of representations of the natural number n as a sum of four fifth powers. In this paper, we establish a non-trivial upper bound for R(n), which is expressed in the following theorem.

It is proved that the variety of all 4-Engel groups of exponent 4 is a maximal proper subvariety of the Burnside variety [Bfr ]4, and the consequences of this are discussed for the finite basis problem for varieties of groups of exponent 4. It is proved that, for r [ges ] 2, the 4-Engel verbal subgroup of the r-generator Burnside group B(r, 4) is irreducible as an [ ]2GL(r, 2)-module. It is observed that the variety of all 4-Engel groups of exponent 4 is insoluble, but not minimal insoluble.

To a reduced plane curve singularity with r branches there is associated its semigroup, a subsemigroup of ℤr[ges ]0. For an irreducible curve singularity, the description of its semigroup in terms of its set of generators is well known. For a reducible curve there exists a combinatorial description of the semigroup. The semigroup of a plane curve singularity with several branches is described in terms of the minimal set of generators constructed in a natural geometric way.

In this paper we describe our discovery that the sporadic simple groups HS and M22 are contained in the simple Chevalley group E7(5).

The work of [9] produces a short list of the possibilities for a sporadic simple subgroup of an exceptional group of Lie type. Apart from possible embeddings of M22 and HS in groups of type E7 in characteristic 5, all of the embeddings of [9] are already known to occur. Thus our paper completes the classification of sporadic simple subgroups of exceptional groups of Lie type.

We give two proofs of the embedding HS<E7(5). The first of these proofs is entirely computer free, while the second proof makes some use of machine calculations. As a step in our hand proof of HS<E7(5) we establish the embedding M22<E7(5): of course, since M22 is a subgroup of HS, this result also follows as a consequence of our computer proof of HS<E7(5).

We were led to conjecture the inclusion HS<E7(5) for the following reasons (similar arguments are presented in [9]). The double cover 2.HS has a faithful 56- dimensional character, whose values are compatible with the character values of groups of type E7 acting on their natural 56-dimensional module. Now HS and 2.HS contain a subgroup 52[ratio ]20, an elementary abelian group of order 25 extended by a cyclic group of order 20 acting faithfully on the 52. Since 20 is not the order of an element in the Weyl group W(E7) = 2×S6(2), it can be shown that 52[ratio ]20 does not embed in groups of type E7(K), where K is a field of characteristic prime to 5. Thus HS embeds in E7(q) only if 5[mid ]q. On the other hand, all local subgroups of 2.HS embed in 2.E7(5), where our conjecture HS<E7(5).

Throughout, G denotes the double cover 2.E7(5), G¯ denotes the simple group E7(5), and V denotes the natural 56-dimensional module for G over GF(5). Most of our notation follows that of the atlas [4]. The two sections of our paper are completely independent.

Thompson's famous theorems on singular values–diagonal elements of the orbit of an n×n matrix A under the action (1) U(n) [otimes ] U(n) where A is complex, (2) SO(n) [otimes ] SO(n), where A is real, (3) O(n) [otimes ] O(n) where A is real are fully examined. Coupled with Kostant's result, the real semi-simple Lie algebra [sfr ][ofr ]n, n yields (2) and hence (3) and the sufficient part (the hard part) of (1). In other words, the curious subtracted term(s) are well explained. Although the diagonal elements corresponding to (1) do not form a convex set in [Copf ]n, the projection of the diagonal elements into ℝn (or iℝn) is convex and the characterization of the projection is related to weak majorization. An elementary proof is given for this hidden convexity result. Equivalent statements in terms of the Hadamard product are also given. The real simple Lie algebra [sfr ][ufr ]n, n shows that such a convexity result fits into the framework of Kostant's result. Convexity properties and torus relations are studied. Thompson's results on the convex hull of matrices (complex or real) with prescribed singular values, as well as Hermitian matrices (real symmetric matrices) with prescribed eigenvalues, are generalized in the context of Lie theory. Also considered are the real simple Lie algebras [sfr ][ofr ]p, q and [sfr ][ofr ]p, q, p < q, which yield the rectangular cases. It is proved that the real part and the imaginary part of the diagonal elements of complex symmetric matrices with prescribed singular values are identical to a convex set in ℝn and the characterization is related to weak majorization. The convex hull of complex symmetric matrices and the convex hull of complex skew symmetric matrices with prescribed singular values are given. Some questions are asked.

The main theorem shows that whenever certain amalgamated products act geometrically on a CAT(0) space, the space has non-locally connected boundary. One can now easily construct a wide variety of examples of one-ended CAT(0) groups with non-locally connected boundary. Applications of this theorem to right-angled Coxeter and Artin groups are considered. In particular, it is shown that the natural CAT(0) space on which a right-angled Artin group acts has locally connected boundary if and only if the group is ℤn for some n.

Consider a standard braid diagram as a three-dimensional figure viewed from the top; what happens when this figure is looked at from the side? Then a new braid can be obtained, and studying the connection between the initial braid and the derived braid so obtained provides both a new simple proof for the existence of the right greedy normal form of positive braids and a geometrical interpretation for the automatic structure of the braid groups.

Let k be an algebraically closed field of characteristic p > 0, and let G be a connected, reductive algebraic group over k. In [8] and [11], conditions on the dimension of rational G modules were seen to imply semisimplicity of these modules. In [8], certain of these conditions were extended to cover the finite groups of Lie type. In this paper, we extend some of the results of [11] to cover these finite Lie type groups. The main such extension is the following result.

There has been interest recently concerning when a left ordered group is locally indicable. Chiswell and Kropholler proved that every left ordered solvable-by-finite group is locally indicable, while Bergman gave examples of left ordered groups which are not locally indicable. This paper proves that every left ordered elementary amenable group is locally indicable. Every solvable-by-finite group is elementary amenable, and every elementary amenable group is amenable. The author leaves it as an open problem as to whether every left ordered amenable group is locally indicable.

In the 1960s, Richard J. Thompson introduced a triple of groups F ⊆ T ⊆ G which, among them, supplied the first examples of infinite, finitely presented, simple groups [14] (see [6] for published details), a technique for constructing an elementary example of a finitely presented group with an unsolvable word problem [12], the universal obstruction to a problem in homotopy theory [8], and the first examples of torsion free groups of type FP∞ and not of type FP [5]. In abstract measure theory, it has been suggested by Geoghegan (see [3] or [9, Question 13]) that F might be a counterexample to the conjecture that any finitely presented group with no non-cyclic free subgroup is amenable (admits a bounded, non-trivial, finitely additive measure on all subsets that is invariant under left multiplication). Recently, F has arisen in the theory of groups of diagrams over semigroup presentations [10], and as the object of questions in the algebra of string rewriting systems [7]. For more extensive bibliographies and more results on Thompson's groups and their generalizations see [1, 4, 6].

A persistent peculiarity of Thompson's groups is their ability to pop up in diverse areas of mathematics. This suggests that there might be something very natural about Thompson's groups. We support this idea by showing (Theorem 1.1 below) that PLo(I), the group of piecewise linear (finitely many changes of slope), orientation-preserving, self-homeomorphisms of the unit interval, is riddled with copies of F: a very weak criterion implies that a subgroup of PLo(I) must contain an isomorphic copy of F.

The starting point of our investigation is the remarkable paper [2] in which Bestvina and Brady gave an example of an infinitely related group of type FP2. The result about right-angled Artin groups behind their example is best interpreted by means of the Bieri–Strebel–Neumann–Renz Σ-invariants.

For a group G the invariants Σn(G) and Σn(G, ℤ) are sets of non-trivial homomorphisms χ[ratio ]G→ℝ. They contain full information about finiteness properties of subgroups of G with abelian factor groups. The main result of [2] determines for the canonical homomorphism χ, taking each generator of the right-angled Artin group G to 1, the maximal n with χ ∈ Σn(G), respectively χ ∈ Σn(G, ℤ).

In [6] Meier, Meinert and VanWyk completed the picture by computing the full Σ-invariants of right-angled Artin groups using as well the result of Bestvina and Brady as algebraic techniques from Σ-theory. Here we offer a new account of their result which is totally geometric. In fact, we return to the Bestvina–Brady construction and simplify their argument considerably by bringing a more general notion of links into play. At the end of the first section we re-prove their main result. By re-computing the full Σ-invariants, we show in the second section that the simplification even adds some power to the method. The criterion we give provides new insight on the geometric nature of the ‘n-domination’ condition employed in [6].

By considering branched coverings of piecewise Euclidean cubical complexes, the paper provides an example of a torsion free hyperbolic group containing a finitely presented subgroup which is not hyperbolic.

Let P be a fixed p-group for an odd prime. We are interested in the localized classifying spaces BG(p) (hereafter denoted simply by BG) for various groups G sharing the same Sylow p-subgroup P. If such groups G become bigger then BG becomes smaller, because a transfer argument shows that BG is a stable summand of BP and of intermediate subgroups. For this reason, C. B. Thomas [16] first found that, in many cases, the odd component of the cohomology of sporadic simple groups should be simple even if the cohomology of P is quite complicated. In particular, when G is the biggest Janko group J4, D. J. Green [6] showed that Heven(G)/3 is the Dickson algebra of rank two. The Janko group has a Sylow 3-subgroup 31+2+ = E, the extra special 3-group of order 33 and of exponent 3. Tezuka and Yagita [14] studied the even-dimensional cohomology of all sporadic simple groups with [mid ]P[mid ] = p3; indeed, in these cases P is isomorphic to the extra-special group p1+2+ = E.

In this paper, we study BG for simple groups G with a Sylow p-subgroup E. We note the importance of G-conjugacy classes of p-pure elementary abelian p-subgroups of rank two. Here ‘p-pure’ means that all nonzero elements in the subgroup are G-conjugate. In Section 1, we see that BG is expressed as a homotopy pushout of B(NG(E)) and of B(p-pure subgroup), when NG(E) = NG(Z(E)). The last condition is always satisfied if p>3. In Section 2, the case p = 3 is studied; for example, we see the homotopy equivalence BJ4 ≅ BRu. In Section 3, we show that stable homotopy splitting is given from the dominant summands of E and of non-p-pure subgroups. When p = 3 the splitting is given explicitly; for example, BJ4 is the summand induced from the trivial representation in F3[Out(E)]. In the last section, we add the list of sporadic simple groups (due to Yoshiara) and cohomology H*(G)/√0 for p = 3 from the paper [14] for the reader's convenience. The author thanks Satoshi Yoshiara, who pointed out to him the paper of Benson and taught him the importance of p-pure subgroups. He also thanks Michishige Tezuka; indeed, most of the results in this paper are natural consequences of results in the joint paper [14] with Tezuka.

It is shown that the division ring of quotients of a skew polynomial ring of automorphism type, if infinite-dimensional over its centre k and satisfying suitable hypotheses, contains the group algebra of a free group of large rank (usually at least [mid ]k[mid ]). The result applies, in particular, to the skew polynomial rings constructed from rational function fields, and affirmatively settles the conjectures of Makar–Limanov and Lichtman in this case.

It is shown that, if Q is a finitely generated abelian group, a finitely generated Q-group A is m-tame if and only if the mth tensor power of the augmentation ideal of ℤA is finitely generated over Am [rtimes ] Q, where Q acts diagonally on both Am and the tensor power. It is proved that quotients of metabelian groups of type FP3 are again of type FP3, and a necessary condition is found for a split extension of abelian-by-(nilpotent of class two) groups to be of type FP2. A conjecture is formulated that generalises the FPm-Conjecture for metabelian groups, and it is shown that one of the implications holds in the prime characteristic case.

A coherent system ([Escr ], V) consists of a holomorphic bundle plus a linear subspace of its space of holomorphic sections. Motivated by the usual notion in geometric invariant theory, a notion of slope stability can be defined for such objects. In the paper it is shown that stability in this sense is equivalent to the existence of solutions to a certain set of gauge theoretic equations. One of the equations is essentially the vortex equation (that is, the Hermitian–Einstein equation with an additional zeroth order term), and the other is an orthonormality condition on a frame for the subspace V⊂H0([Escr ]).

A minimax principle is derived for the eigenvalues in the spectral gap of a possibly non-semibounded self-adjoint operator. It allows the nth eigenvalue of the Dirac operator with Coulomb potential from below to be bound by the nth eigenvalue of a semibounded Hamiltonian which is of interest in the context of stability of matter. As a second application it is shown that the Dirac operator with suitable non-positive potential has at least as many discrete eigenvalues as the Schrödinger operator with the same potential.

The special linear group is the simply connected group and the projective linear group is the adjoint group of Lie type An. They are distinguished sections of the (reductive) general linear group which certainly is of this type as well (root system). We shall characterize the general linear group as the universal group of type An. Indeed we shall introduce corresponding algebraic groups and finite groups for each Lie type (to indecomposable root systems). Knowledge of the universal group implies knowledge of the related simply connected and adjoint groups; in certain respects the universal group even appears to be better behaved (automorphisms, Schur multiplier, character table).

The paper considers nonlinear, probability measure-valued dynamical systems that generalise those classical Lotka–Volterra equations in which, to use ecological terminology, the total size of a finite number of populations of interacting species is conserved. In this generalisation there is a ‘different species’ at each point of an arbitrary measurable space.

Such infinitely-many-species analogues of the classical Lotka–Volterra equations appear as hydrodynamic-type limits of stochastic systems of randomly coalescing masses related to those that have been used to model physical and chemical processes of agglomeration, coagulation and condensation.

One natural instance of this generalisation has closed-form solutions, including a family of solutions that exhibit soliton-like behaviour. The large time asymptotics of other classes of examples can be completely described using analogues of Lyapunov function techniques. Moreover, there are conserved quantities in the form of relative entropies that generalise those found by Volterra in the classical case. Finally, each solution has a series expansion as a time-varying, geometric mixture of a fixed sequence of probability measures. The existence of this expansion is related to the fact that the system is in martingale problem duality with a function-valued Markov process.

The paper proves that a set which contains spheres centered at all points of a set of Hausdorff dimension greater than 1 must have positive Lebesgue measure. It also proves the corresponding result for circles, provided that the set of centers has Hausdorff dimension greater than 3/2.