Let (Gi | i ∈ I) be a family of groups, let F be a free group, and let the free product of F and all the Gi.

Let denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, By the Kurosh Subgroup Theorem, every element of is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as To avoid the vacuous case, we make the additional assumption that contains a non-cyclic group, and we defineWe are interested in precise bounds for . In the special case where I is empty, Hanna Neumann proved that ∈ [1,2], and conjectured that = 1; fifty years later, this interval has not been reduced.

With the understanding that ∞/(∞ − 2) is 1, we define

Generalizing Hanna Neumann's theorem we prove that , and, moreover, whenever G has 2-torsion. Since is finite, is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that whenever G does not have 2-torsion.

We construct a pairing on the dual Selmer group over false Tate curve extensions of an elliptic curve with good ordinary reduction at a prime p≥5. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of the p-adic L-function. As an application we compute the characteristic elements of those modules – arising naturally in the Iwasawa-theory for elliptic curves over the false Tate curve extension – which have rank 1 over the subgroup of the Galois group fixing the cyclotomic extension of the ground field. We also show that the example of a non-principal reflexive left ideal of the Iwasawa algebra does not rule out the possibility that all torsion Iwasawa-modules are pseudo-isomorphic to the direct sum of quotients of the algebra by principal ideals.

We study H*(P), the mod p cohomology of a finite p-group P, viewed as an –module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e ∈ is a nonzero idempotent, then the Krull dimension of eH*(P) equals the rank of P. We prove this for all p-groups when p is odd, and for many 2–groups.

This paper discusses the Brauer–Manin obstruction on double covers of the projective plane branched along a plane section of a Kummer surface from both the practical and the theoretical points of view. Theoretical highlights include the determination of a complete set of generators for the Brauer group; on the practical side, we give several surfaces with a Brauer–Manin obstruction and verify that the Brauer–Manin obstruction is the only one for a collection of several thousand surfaces of this type.

We establish isomorphisms between certain specializations of BMW algebras and the symmetric squares of Temperley–Lieb algebras. These isomorphisms imply a link-polynomial identity due to W. B. R. Lickorish. As an application, we compute the closed images of the irreducible braid group representations factoring over these specialized BMW algebras.

Any 2-bridge knot in S3 has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.

Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmüller space. The main tool for these results is the known mapping class group invariant ideal cell decomposition of the Teichmüller space.

This new 1-cocycle is mapping class group equivariant, so various contractions of its powers yield various combinatorial (co)cycles of the moduli space of curves, which are also new. Our combinatorial construction can be related to former works of Kawazumi and the first-named author with the consequence that the algebra generated by the cohomology classes represented by the new cocycles is precisely the tautological algebra of the moduli space.

There is finally a discussion of prospects for similarly finding cocycle lifts of the higher Johnson homomorphisms.

We observe that Cheng's Eigenvalue Comparison Theorem for normal geodesic balls [4] is still valid if we impose bounds on the mean curvature of the distance spheres instead of bounds on the sectional and Ricci curvatures. In this version, there is a weak form of rigidity in case of equality of the eigenvalues. Namely, equality of the eigenvalues implies that the distance spheres of the same radius on each ball has the same mean curvature. On the other hand, we construct smooth metrics , non-isometric to the standard metric canκ of constant sectional curvature κ, such that the geodesic balls have the same first eigenvalue, the same volume and the distance spheres and, have the same mean curvatures. In the end, we apply this version of Cheng's Eigenvalue Comparison Theorem to construct examples of Riemannian manifolds M with arbitrary topology with positive fundamental tone λ*(M)>0 extending Veeravalli's examples,[7]

We prove that for an action of a finite group G on a systolic complex X there exists a G–invariant subcomplex of X of diameter ≤5. For 7–systolic locally finite complexes we prove there is a fixed point for the action of any finite G. This implies that free products with amalgamation (and HNN extensions) of 7–systolic groups over finite subgroups are also 7–systolic.

We introduce the notion of character amenable Banach algebras. We prove that character amenability for either of the group algebra L1(G) or the Fourier algebra A(G) is equivalent to the amenability of the underlying group G. Character amenability of the measure algebra M(G) is shown to be equivalent to G being a discrete amenable group. We also study functorial properties of character amenability. For a commutative character amenable Banach algebra A, we prove all cohomological groups with coefficients in finite-dimensional Banach A-bimodules, vanish. As a corollary we conclude that all finite-dimensional extensions of commutative character amenable Banach algebras split strongly.

This paper investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masseswhere (xn)n≥0 is a sequence of points in [0, 1]d and (wn)n≥0 is a positive sequence of weights such that Σn≥0wn < ∞. We consider the case where the points xn are roughly uniformly distributed in [0, 1]d, and the weights wn depend on a random self-similar measure μ, a parameter ρ ∈ (0, 1], and a sequence of positive radii (λn)n≥1 converging to 0 in the following wayThe measure ν has a rich multiscale structure. The computation of its multifractal spectrum is related to heterogeneous ubiquity properties of the system {(xn,λn)n with respect to μ.

The rolling sphere problem on Euclidean space consists of determining the path of minimal length traced by the point of contact of the oriented unit sphere as it rolls on without slipping between two points of . This problem is extended to situations in which an oriented sphere of radius ρ rolls on a stationary sphere and to the hyperbolic analogue in which the spheres and are replaced by the hyperboloids and respectively. The notion of “rolling” is defined in an isometric sense: the length of the path traced by the point of contact is measured by the Riemannian metric of the stationary manifold, and the orientation of the rolling object is measured by a matrix in its isometry group. These rolling problems are formulated as left invariant optimal control problems on Lie groups whose Hamiltonian extremal equations reveal two remarkable facts: on the level of Lie algebras the extremal equations of all these rolling problems are governed by a single set of equations, and the projections onto the stationary manifold of the extremal equations having I4=0, where I4 is an integral of motion, coincide with the elastic curves on this manifold. The paper then outlines some explicit solutions based on the use of symmetries and the corresponding integrals of motion.

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3–manifold is diffeo morphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3–manifolds that cannot arise as a cusp cross-section of a 1–cusped nonsingular Hilbert modular surface.

In this paper, the dates when the paper and its revised version were received, were inadvertently omitted. The missing information is: