The Ore condition, affiliated operators, and the lamplighter group

Let G be the wreath product of Z and Z/2, the so called lamplighter group and k a commutative ring. We show that kG does not have a classical ring of quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka notebook problem. Assume that kG is contained in a ring R in which the element 1-x is invertible, with x a generator of Z considered as subset of G. Then R is not flat over kG. If k is the field of complex numbers, this applies in particular to the algebra UG of unbounded operators affiliated to the group von Neumann algebra of G. We present two proofs of these results. The second one is due to Warren Dicks, who, having seen our argument, found a much simpler and more elementary proof, which at the same time yielded a more general result than we had originally proved. Nevertheless, we present both proofs here, in the hope that the original arguments might be of use in some other context not yet known to us.Comment: LaTex2e, 7 pages. Added a new proof of the main result (due to Warren Dicks) which is shorter, easier and more elementary, and at the same time yields a slightly more general result. Additionally: misprints removed. to appear in Proceedings of "Higher dimensional manifold theory", Conference at ICTP Trieste 200

Analysis of a four-mirror cavity enhanced Michelson interferometer

We investigate the shot noise limited sensitivity of a four-mirror cavity enhanced Michelson interferometer. The intention of this interferometer topology is the reduction of thermal lensing and the impact of the interferometers contrast although transmissive optics are used with high circulating powers. The analytical expressions describing the light fields and the frequency response are derived. Although the parameter space has 11 dimensions, a detailed analysis of the resonance feature gives boundary conditions allowing systematic parameter studies

Optimal time-domain combination of the two calibrated output quadratures of GEO 600

GEO 600 is an interferometric gravitational wave detector with a 600 m arm-length and which uses a dual-recycled optical configuration to give enhanced sensitivity over certain frequencies in the detection band. Due to the dual-recycling, GEO 600 has two main output signals, both of which potentially contain gravitational wave signals. These two outputs are calibrated to strain using a time-domain method. In order to simplify the analysis of the GEO 600 data set, it is desirable to combine these two calibrated outputs to form a single strain signal that has optimal signal-to-noise ratio across the detection band. This paper describes a time-domain method for doing this combination. The method presented is similar to one developed for optimally combining the outputs of two colocated gravitational wave detectors. In the scheme presented in this paper, some simplifications are made to allow its implementation using time-domain methods

Superconductor-Ferromagnet Bi-Layers: a Comparison of s-Wave and d-Wave Order Parameters

We study superconductor-ferromagnet bi-layers, not only for s-wave but also for d-wave superconductors. We observe oscillations of the critical temperature when varying the thickness of the ferromagnetic layer for both s-wave and d-wave superconductors. However, for a rotated d-wave order parameter the critical temperature differs considerably from that for the unrotated case. In addition we calculate the density of states for different thicknesses of the ferromagnetic layer; the results reflect the oscillatory behaviour of the superconducting correlations.Comment: 11 pages, 5 figures, accepted for publication in J. Phys.: Condens. Matte

The universal functorial equivariant Lefschetz invariant

We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K_0 of the category of "phi-endomorphisms of finitely generated free RPi(G,X)-modules". We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds.Comment: 33 pages; shortened version of the author's PhD thesis, supervised by Wolfgang Lueck, Westfaelische Wilhelms-Universitaet Muenster, 200

Thermal noise of folding mirrors

Current gravitational wave detectors rely on the use of Michelson interferometers. One crucial limitation of their sensitivity is the thermal noise of their optical components. Thus, for example fluctuational deformations of the mirror surface are probed by a laser beam being reflected from the mirrors at normal incidence. Thermal noise models are well evolved for that case but mainly restricted to single reflections. In this work we present the effect of two consecutive reflections under a non-normal incidence onto mirror thermal noise. This situation is inherent to detectors using a geometrical folding scheme such as GEO\,600. We revise in detail the conventional direct noise analysis scheme to the situation of non-normal incidence allowing for a modified weighting funtion of mirror fluctuations. An application of these results to the GEO\,600 folding mirror for Brownian, thermoelastic and thermorefractive noise yields an increase of displacement noise amplitude by 20\% for most noise processes. The amplitude of thermoelastic substrate noise is increased by a factor 4 due to the modified weighting function. Thus the consideration of the correct weighting scheme can drastically alter the noise predictions and demands special care in any thermal noise design process

Surgery groups of the fundamental groups of hyperplane arrangement complements

Using a recent result of Bartels and Lueck (arXiv:0901.0442) we deduce that the Farrell-Jones Fibered Isomorphism conjecture in L-theory is true for any group which contains a finite index strongly poly-free normal subgroup, in particular, for the Artin full braid groups. As a consequence we explicitly compute the surgery groups of the Artin pure braid groups. This is obtained as a corollary to a computation of the surgery groups of a more general class of groups, namely for the fundamental group of the complement of any fiber-type hyperplane arrangement in the complex n-space.Comment: 11 pages, AMSLATEX file, revised following referee's comments and suggestions, to appear in Archiv der Mathemati

On Turing dynamical systems and the Atiyah problem

Main theorems of the article concern the problem of M. Atiyah on possible values of l^2-Betti numbers. It is shown that all non-negative real numbers are l^2-Betti numbers, and that "many" (for example all non-negative algebraic) real numbers are l^2-Betti numbers of simply connected manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to a simply connected manifold with a transcendental l^2-Betti number with respect to an action of the threefold direct product of the lamplighter group Z/2 wr Z. The main new idea is embedding Turing machines into integral group rings. The main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids.Comment: 35 pages; essentially identical to the published versio

Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory

We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H-flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which provides new consistency conditions for fractional D-branes and Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from equivariant K-theory to Bredon cohomology to define new Ramond-Ramond couplings of D-branes which generalize previous examples. We propose a definition for groups of differential characters associated to equivariant K-theory. We derive a Dirac quantization rule for Ramond-Ramond fluxes, and study flat Ramond-Ramond potentials on orbifolds.Comment: 46 pages; v2: typos correcte

Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-M\"uller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L2 analytic and Reidemeister torsions are equal.Comment: 78 pages, AMSTe