Let q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. It is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. (These special situations are eliminated in a companion paper by the first author.) An example coming from the exceptional group E6(q) shows that the kernel of the action of Gx on the ball of radius 2 around x can be non-trivial.

Let q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. In a companion paper, it is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. In this paper we show that these special situations cannot occur.

Let f(x) ∈ [x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1/r1, r2, 1/r2, . . ., rd, 1/rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 + 1/r1, . . ., rd + 1/rd. If the Galois groups of f and g are Gf and Gg respectively, then Gg ≅ Gf/N, where N is isomorphic to a subgroup of C2d. In a naive sense, the generic case is Gf ≅ C2d ⋊ Sd, with N ≅ C2d and Gg ≅ Sd. When f(x) has extra structure this may be reflected in the Galois group, and it is not always true even that Gf ≅ N ⋊ Gg. For example, for cyclotomic polynomials f(x) = Φn(x) it is known that Gf ≅ N ⋊ Gg if and only if n is divisible either by 4 or by some prime congruent to 3 modulo 4.

In this paper we deal with irreducible reciprocal monic polynomials f(x) ∈ [x] that are ‘close’ to being cyclotomic, in that there is one pair of real positive reciprocal roots and all other roots lie on the unit circle. With the further restriction that f(x) has degree at least 4, this means that f(x) is the minimal polynomial of a Salem number. We show that in this case one always has Gf ≅ N ⋊ Gg, and moreover that N ≅ C2d or C2d−1, with the latter only possible if d is odd.

We study generalized symbolic powers and form ideals of powers and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated.

Let p be a prime number and M a quadratic number field, M ≠ ℚ() if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/ℚ with Galois group D2p and an elliptic curve E/ℚ such that F contains M and the p-Selmer group of E/F has size at least pd.

We give an elementary construction of the compact real form of the Lie algebra 2. This construction exhibits the group 23·L3(2) as a group of automorphisms. We also show that there is a unique 14-dimensional real Lie algebra invariant under the action of this group.

For a real abelian number field F and for a prime p we study the relation between the p-parts of the class groups and of the quotients of global units modulo cyclotomic units along the cyclotomic p-extension of F. Assuming Greenberg's conjecture about the vanishing of the λ-invariant of the extension, a map between these groups has been constructed by several authors, and shown to be an isomorphism if p does not split in F. We focus in the split case, showing that there are, in general, non-trivial kernels and cokernels.

We illustrate a minor error in the biadjointness result for 2-categories of traced monoidal categories and tortile monoidal categories stated by Joyal, Street and Verity. We also show that the biadjointness holds after suitably changing the definition of 2-cells.

The unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this paper we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equals its unknotting number, if and only if the given knot appears in an unknotting sequence of a torus knot.

Here we determine the arithmetic data i.e. the totally real number field and the set of ramified places of the defining quaternion algebra, of all those commensurability classes of arithmetic Fuchsian groups which contain a surface group of genus 2, i.e. a group of signature (2;– –).

A Klein–Schottky group is an extended Kleinian group, containing no reflections and whose orientation-preserving half is a Schottky group. A dihedral-Klein–Schottky group is an extended Kleinian group generated by two different Klein–Schottky groups, both with the same orientation-preserving half. We provide a structural description of the dihedral-Klein–Schottky groups.

Let M be a handlebody of genus g, with a Schottky structure. An imaginary reflection τ of M is an orientation-reversing homeomorphism of M, of order two, whose restriction to its interior is an hyperbolic isometry having at most isolated fixed points. It is known that the number of fixed points of τ is at most g + 1; τ is called a maximal imaginary reflection if it has g + 1 fixed points. As a consequence of the structural description of the dihedral-Klein–Schottky groups, we are able to provide upper bounds for the cardinality of the set of fixed points of two or three different imaginary reflections acting on a handlebody with a Schottky structure. In particular, we show that maximal imaginary reflections are unique.

Multisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.

We consider sets of real numbers in [0, 1) with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by 1/2, are given by a modified variational principle.