Proceedings of the London Mathematical Society - current issue

Artin braid groups and homotopy groups

Li, J., Wu, J. — Mon, 19 Oct 2009 08:26:28 PDT

We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as summands of the centres of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. The results give new connections between the braid groups and the general higher homotopy groups of spheres.

The curvature invariant for a class of homogeneous operators

Misra, G., Shyam Roy, S. — Mon, 19 Oct 2009 08:26:28 PDT

For an operator T in the class B_n(), introduced by Cowen and Douglas, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle E_T determine the unitary equivalence class of the operator T. In a subsequent paper, the authors ask if the simultaneous unitary equivalence class of the curvature and these covariant derivatives are necessary to determine the unitary equivalence class of the operator T B_n(). Here we show that some of the covariant derivatives are necessary. Our examples consist of homogeneous operators in B_n(D). For homogeneous operators, the simultaneous unitary equivalence class of the curvature and all its covariant derivatives at any point w in the unit disc D are determined from the simultaneous unitary equivalence class at 0. This shows that it is enough to calculate all the invariants and compare them at just one point, say 0. These calculations are then carried out in number of examples. One of our main results is that the curvature along with its covariant derivative of order (0, 1) at 0 determines the equivalence class of generic homogeneous Hermitian holomorphic vector bundles over the unit disc.

Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup

Bludov, V. V., Glass, A. M. W. — Mon, 19 Oct 2009 08:26:29 PDT

We use permutation groups to give necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable. We obtain several consequences answering previously posed problems and also prove the right-orderable analogues of the Higman Embedding Theorem and the Boone–Higman Theorem.

Cohomogeneity one disk bundles with normal homogeneous collars

Schwachhofer, L. J., Tapp, K. — Mon, 19 Oct 2009 08:26:29 PDT

We consider cohomogeneity one homogeneous disk bundles and address the question when these admit a nonnegatively curved invariant metric with normal collar, that is, such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in {2, 3, 4, 6, 8}. Moreover, we give a complete classification of such bundles of rank 6 and 8, and a partial classification for rank 3.

Solution of the polynomial moment problem

Pakovich, F., Muzychuk, M. — Mon, 19 Oct 2009 08:26:29 PDT

In this paper we give a solution of the following ‘polynomial moment problem’ which arose about 10 years ago in connection with Poincaré's center-focus problem: for a given polynomial P(z) to describe polynomials q(z) orthogonal to all powers of P(z) on a segment [a, b].

On the self-similarity problem for ergodic flows

Fraczek, K., Lemanczyk, M. — Mon, 19 Oct 2009 08:26:30 PDT

Given an ergodic flow (T_t)_{t R} we study the problem of its self-similarities, that is, we want to describe the set of s R for which the original flow is isomorphic to the flow (T_st)_{t R}. The problem is examined in some classes of special flows over irrational rotations and over interval exchange transformations. In particular, translation flows on translation surfaces are considered: we prove that under the weak mixing condition the set of self-similarities has Lebesgue measure zero. For von Neumann special flows over irrational rotations given by Diophantine numbers, this set is shown to be equal to {1}, while for horocycle flows a weak convergence in case of some singular (with respect to the volume measure) measures is shown to give rise to some new equidistribution result. The problem of self-similarity is also studied from the spectral point of view, especially in the class of Gaussian systems.

Geometric criteria for Landweber exactness

Hollander, S. — Mon, 19 Oct 2009 08:26:30 PDT

The purpose of this paper is to give a new presentation of some of the main results concerning Landweber exactness in the context of the homotopy theory of stacks. We present two new criteria for Landweber exactness over a flat Hopf algebroid. The first criterion is used to classify stacks arising from Landweber exact maps of rings. Using as extra input only Lazard's theorem and Cartier's classification of p-typical formal group laws, this result is then applied to deduce many of the main results concerning Landweber exactness in stable homotopy theory and to compute the Bousfield classes of certain BP-algebra spectra. The second criterion can be regarded as a generalization of the Landweber exact functor theorem, and we use it to give a proof of the original theorem.

Building blocks of etale endomorphisms of complex projective manifolds

Nakayama, N., Zhang, D.-Q. — Mon, 19 Oct 2009 08:26:30 PDT

Étale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true. They are the endomorphisms of abelian varieties and the nearly étale rational endomorphisms of weak Calabi–Yau varieties.

A solution to the Douglas-Rudin problem for matrix-valued functions

Barclay, S. — Mon, 19 Oct 2009 08:26:31 PDT

We solve the noncommutative Douglas–Rudin problem, showing that any log-integrable essentially bounded square matrix-valued function f can be written in the form h*g, where h and g lie in H . Extensions to other L ^p spaces, with norm bounds on the factors of f, are also provided.

The effect of convolving families of L-functions on the underlying group symmetries

Duenez, E., Miller, S. J. — Mon, 19 Oct 2009 08:26:31 PDT

Let {F_N} and {G_M} be families of primitive automorphic L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N, M -> , the statistical behavior (1-level density) of the low-lying zeros of L-functions in F_N and G_M agrees with that of the eigenvalues near 1 of matrices in G₁ and G₂, respectively, as the size of the matrices tend to infinity, where each G_i is one of the classical compact groups (unitary U, symplectic Sp, or orthogonal O, SO(even), SO(odd)). Assuming that the convolved families of L-functions F_N x G_M are automorphic, we study their 1-level density. (We also study convolved families of the form f x G_M for a fixed f.) Under natural assumptions on the families (which hold in many cases), we can associate to each family L of L-functions a symmetry constant cL equal to 0, 1, or–1 if the corresponding low-lying zero statistics agree with those of the unitary symplectic, or orthogonal group, respectively. Our main result is that cFxG=cF·cG: the symmetry type of the convolved family is the product of the symmetry types of the two families. A similar statement holds for the convolved families f x G_M. We provide examples built from Dirichlet L-functions and holomorphic modular forms and their symmetric powers. An interesting special case is to convolve two families of elliptic curves with positive rank. In this case the symmetry group of the convolution is independent of the ranks, in accordance with the general principle of multiplicativity of the symmetry constants (but the ranks persist, before taking the limit N, M -> , as lower-order terms).