The definability criterions for convex projective polyhedral reflection groups

Following Vinberg, we find the criterions for a subgroup generated by reflections \Gamma \subset \SL^{\pm}(n+1,\mathbb{R}) and its finite-index subgroups to be definable over $\mathbb{A}$ where $\mathbb{A}$ is an integrally closed Noetherian ring in the field $\mathbb{R}$. We apply the criterions for groups generated by reflections that act cocompactly on irreducible properly convex open subdomains of the $n$-dimensional projective sphere. This gives a method for constructing injective group homomorphisms from such Coxeter groups to \SL^{\pm}(n+1,\mathbb{Z}). Finally we provide some examples of \SL^{\pm}(n+1,\mathbb{Z})-representations of such Coxeter groups. In particular, we consider simplicial reflection groups that are isomorphic to hyperbolic simplicial groups and classify all the conjugacy classes of the reflection subgroups in \SL^{\pm}(n+1,\mathbb{R}) that are definable over $\mathbb{Z}$. These were known by Goldman, Benoist, and so on previously.Comment: 31 pages, 8 figure

On compatibility between isogenies and polarisations of abelian varieties

We discuss the notion of polarised isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarisations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarised isogenies can be reduced to questions about unpolarised isogenies or vice versa. Our main theorem concerns abelian varieties B which are isogenous to a fixed abelian variety A. It establishes the existence of a polarised isogeny A to B whose degree is polynomially bounded in n, if there exist both an unpolarised isogeny A to B of degree n and a polarised isogeny A to B of unknown degree. As a further result, we prove that given any two principally polarised abelian varieties related by an unpolarised isogeny, there exists a polarised isogeny between their fourth powers. The proofs of both theorems involve calculations in the endomorphism algebras of the abelian varieties, using the Albert classification of these endomorphism algebras and the classification of Hermitian forms over division algebras

Compact pseudo-Riemannian manifolds with parallel Weyl tensor

It is shown that in every dimension n=3j+2, j=1,2,3,..., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial torus bundles over the circle. They all arise from a construction that a priori yields bundles over the circle, having as the fibre either a torus, or a 2-step nilmanifold with a complete flat torsionfree connection; our argument only realizes the torus case.Comment: 19 page

Anosov representations: Domains of discontinuity and applications

The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher Teichmueller spaces, and for lattices in SO(1,n). In this article we extend the notion of Anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties. In particular, given an Anosov representation of $\Gamma$ into G we explicitly construct open subsets of compact G-spaces, on which $\Gamma$ acts properly discontinuously and with compact quotient. As a consequence we show that higher Teichmueller spaces parametrize locally homogeneous geometric structures on compact manifolds. We also obtain applications regarding (non-standard) compact Clifford-Klein forms and compactifications of locally symmetric spaces of infinite volume.Comment: 63 pages, accepted for publication in Inventiones Mathematica

Lack of Foxp3 function and expression in the thymic epithelium

Foxp3 is essential for the commitment of differentiating thymocytes to the regulatory CD4+ T (T reg) cell lineage. In humans and mice with a genetic Foxp3 deficiency, absence of this critical T reg cell population was suggested to be responsible for the severe autoimmune lesions. Recently, it has been proposed that in addition to T reg cells, Foxp3 is also expressed in thymic epithelial cells where it is involved in regulation of early thymocyte differentiation and is required to prevent autoimmunity. Here, we used genetic tools to demonstrate that the thymic epithelium does not express Foxp3. Furthermore, we formally showed that genetic abatement of Foxp3 in the hematopoietic compartment, i.e. in T cells, is both necessary and sufficient to induce the autoimmune lesions associated with Foxp3 loss. In contrast, deletion of a conditional Foxp3 allele in thymic epithelial cells did not result in detectable changes in thymocyte differentiation or pathology. Therefore, in mice the only known role for Foxp3 remains promotion of T reg cell differentiation within the T cell lineage, whereas there is no role for Foxp3 in thymic epithelial cells

The XXL Survey VII: A supercluster of galaxies at z=0.43

The XXL Survey is the largest homogeneous and contiguous survey carried out with XMM-Newton. Covering an area of 50 square degrees distributed over two fields, it primarily investigates the large-scale structures of the Universe using the distribution of galaxy clusters and active galactic nuclei as tracers of the matter distribution. Given its depth and sky coverage, XXL is particularly suited to systematically unveiling the clustering of X-ray clusters and to identifying superstructures in a homogeneous X-ray sample down to the typical mass scale of a local massive cluster. A friends-of-friends algorithm in three-dimensional physical space was run to identify large-scale structures. In this paper we report the discovery of the highest redshift supercluster of galaxies found in the XXL Survey. We describe the X-ray properties of the clusters members of the structure and the optical follow-up. The newly discovered supercluster is composed of six clusters of galaxies at a median redshift z around 0.43 and distributed across approximately 30 by 15 arc minutes (10 by 5 Mpc on sky) on the sky. This structure is very compact with all the clusters residing in one XMM pointing; for this reason this is the first supercluster discovered with the XXL Survey. Spectroscopic follow-up with WHT (William Herschel Telescope) and NTT (New Technology Telescope) confirmed a median redshift of z = 0.43. An estimate of the X-ray mass and luminosity of this supercluster and of its total gas mass put XLSSC-e at the average mass range of superclusters; its appearance, with two members of equal size, is quite unusual with respect to other superclusters and provides a unique view of the formation process of a massive structure.Comment: A&A, accepted; special XXL issu

Flux Confinement in Mesoscopic Superconductors

We report on flux confinement effects in superconducting submicron line, loop and dot structures. The main idea of our study was to vary the boundary conditions for confinement of the superconducting condensate by taking samples of different topology and, through that, modifying the lowest Landau level E_{LLL}(H). Since the critical temperature versus applied magnetic field T_{c}(H) is, in fact, E_{LLL}(H) measured in temperature units, it is varied as well when the sample topology is changed. We demonstrate that in all studied submicron structures the shape of the T_{c}(H) phase boundary is determined by the confinement topology in a unique way.Comment: 10 pages, 5 EPS figures, uses LaTeX's sup.sty, contribution to a special issue of "Superlattices and Microstructures

Semitoric integrable systems on symplectic 4-manifolds

Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce