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We discuss equivariance for linear liftings of measurable functions. Existence is established when a transformation group acts amenably, as e.g. the Moebius group of the projective line. Since the general proof is very simple but not explicit, we also provide a much more explicit lifting for semi-simple Lie groups acting on their Furstenberg boundary, using unrestricted Fatou convergence. This setting is relevant to cocycles for characteristic classes., Comment: Removed the sigma-compactness assumption from Theorem A; minor corrections

Market trade-routes can support infectious-disease transmission, impacting biological populations and even disrupting causal trade. Epidemiological models increasingly account for reductions in infectious contact, such as risk-aversion behaviour in response to pathogen outbreaks. However, market dynamics clearly differ from simple risk-aversion, as are driven by different motivation and conditioned by trade constraints, known in economics as friction, that arise because exchanges are costly. Here we develop a novel economic-market model where transient and long-term market dynamics are determined by trade friction and agent adaptation, and can influence disease transmission. We specify the participants, frequency, volume, and price in trade transactions, and investigate, using analytical insights and simulation, how trade friction affects joint market and epidemiological dynamics. The friction values explored encompass estimates from French cattle and pig markets. We show that, when trade is the dominant route of transmission, market friction can be a significantly stronger determinant of epidemics than risk-aversion behaviour. In particular, there is a critical friction level above which epidemics do not occur. For a given level of friction, open unregulated markets can boost epidemics compared with closed or tightly regulated markets. Our results are robust to model specificities and can hold in the presence of non-trade disease-transmission routes. In particular, we try to explain why outbreaks in French livestock markets appear more frequently in cattle than swine despite swine trade-flow being larger. To minimize contagion in markets, safety policies could generate incentives for larger-volume, less-frequent transactions, increasing trade friction without necessarily affecting overall trade flow., Comment: 51 pages, 9 figures

We introduce a relative fixed point property for subgroups of a locally compact group, which we call relative amenability. It is a priori weaker than amenability. We establish equivalent conditions, related among others to a problem studied by Reiter in 1968. We record a solution to Reiter's problem. We study the class X of groups in which relative amenability is equivalent to amenability for all closed subgroups; we prove that X contains all familiar groups. Actually, no group is known to lie outside X. Since relative amenability is closed under Chabauty limits, it follows that any Chabauty limit of amenable subgroups remains amenable if the ambient group belongs to the vast class X., Comment: We added a solution to Reiter's problem and a discussion of L^1-equivariance

Supramenability of groups is characterised in terms of invariant measures on locally compact spaces. This opens the door to constructing interesting crossed product C*-algebras for non-supramenable groups. In particular, stable Kirchberg algebras in the UCT class are constructed using crossed products for both amenable and non-amenable groups., Comment: Minor changes; to appear in Doc. Math

In their 1993 paper, Arak, Clifford and Surgailis discussed a new model of random planar graph. As a particular case, that model yields tessellations with only T-vertices (T-tessellations). Using a similar approach involving Poisson lines, a new model of random T-tessellations is proposed. Campbell measures, Papangelou kernels and Georgii-Nguyen-Zessin formulae are translated from point process theory to random T-tessellations. It is shown that the new model shows properties similar to the Poisson point process and can therefore be considered as a completely random T-tessellation. Gibbs variants are introduced leading to models of random T-tessellations where selected features are controlled. Gibbs random T-tessellations are expected to better represent observed tessellations. As numerical experiments are a key tool for investigating Gibbs models, we derive a simulation algorithm of the Metropolis-Hastings-Green family.

The group of piecewise projective homeomorphisms of the line provides straightforward counter-examples to the so-called von Neumann conjecture. The examples are so simple that many additional properties can be established., Comment: This version submitted to PNAS on October 22, 2012. Final version published in PNAS at http://dx.doi.org/10.1073/pnas.1218426110

For any group $G$ containing an infinite elementary amenable subgroup, and any $2

It is a well-known open problem since the 1970s whether a finitely generated perfect group can be normally generated by a single element or not. We prove that the topological version of this problem has an affirmative answer as long as we exclude infinite discrete quotients (which is probably a necessary restriction)., Comment: 4 pages (added an example)

We studied magnetic-field induced microwave absorption in 100-200 nm thick La$_{0.7}$Sr$_{0.3}$MnO$_{3}$ films on SrTiO$_{3}$ substrate and found a low-field absorption with a very peculiar angular dependence: it appears only in the oblique field and is absent both in the parallel and in the perpendicular orientations. We demonstrate that this low-field absorption results from the ferromagnetic resonance in the multidomain state (domain-mode resonance). Its unusual angular dependence arises from the interplay between the parallel component of the magnetic field that drives the film into multidomain state and the perpendicular field component that controls the domain width through its effect on domain wall energy. The low-field microwave absorption in the multidomain state can be a tool to probe domain structure in magnetic films with in-plane magnetization., Comment: 9 pages, 9 Figures

We provide the first examples of finitely generated simple groups that are amenable (and infinite). This follows from a general existence result on invariant states for piecewise-translations of the integers. The states are obtained by constructing a suitable family of densities on the classical Bernoulli space., Comment: The proof has been streamlined: the commutator of W(Z) is not needed anymore

Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the torsion-free case. We establish Levi decompositions for stabilisers of points at infinity of X, generalising the case of linear algebraic groups to Is(X). A geometric counterpart of this sheds light on the refined bordification of X (\`a la Karpelevich) and leads to a converse to the Adams-Ballmann theorem. It is further deduced that unimodular cocompact groups cannot fix any point at infinity except in the Euclidean factor; this fact is needed for the study of CAT(0) lattices. Various fixed point results are derived as illustrations., Comment: 33 pages

The magnetic properties of the alkaline earth oxocuprate BaCu3O4 are investigated. We show that the characteristic Cu3O4 layers of this material can be described with diamond chains of antiferromagnetically coupled Cu 1/2 spins with only a weak coupling between two adjacent chains. These Cu3O4 layers seem to represent a so far unique system of weakly coupled one-dimensional magnetic objects where the local AF ordering of the Cu2+ ions leads to an actual net magnetic moment of an isolated diamond chain. We demonstrate a magnetic transition at a high N\'eel temperature T_{N}=336 K.

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic locally compact groups with a cusp-uniform non-uniform lattice, is very restricted., Comment: 41 pages, no figure. v2: revised version (minor changes)

On the one hand, we construct a continuous family of non-isometric proper CAT(-1) spaces on which the isometry group ${\rm Isom}(\mathbf{H}^{n})$ of the real hyperbolic $n$-space acts minimally and cocompactly. This provides the first examples of non-standard CAT(0) model spaces for simple Lie groups. On the other hand, we classify all continuous non-elementary actions of ${\rm Isom}(\mathbf{H}^{n})$ on the infinite-dimensional real hyperbolic space. It turns out that they are in correspondence with the exotic model spaces that we construct., Comment: 42 pages, minor modifications, this is the final version

Grigorchuk and Medynets recently announced that the topological full group of a minimal Cantor Z-action is amenable. They asked whether the statement holds for all minimal Cantor actions of general amenable groups as well. We answer in the negative by producing a minimal Cantor Z^2-action for which the topological full group contains a non-abelian free group., Comment: 3 pages (some comments added)

Recall that a rng is a ring which is possibly non-unital. In this note, we address the problem whether every finitely generated idempotent rng (abbreviated as irng) is singly generated as an ideal. It is well-known that it is the case for a commutative irng. We prove here it is also the case for a free rng on finitely many idempotents and for a finite irng. A relation to the Wiegold problem for perfect groups is discussed., Comment: 5 pages, no figures

We prove a fixed point theorem for a family of Banach spaces, notably L^1 and its non-commutative analogues. Several applications are given, e.g. the optimal solution to the "derivation problem" studied since the 1960s., Comment: minor additions; to appear in Inv. Math

We prove that the norm of the Euler class E for flat vector bundles is $2^{-n}$ (in even dimension $n$, since it vanishes in odd dimension). This shows that the Sullivan--Smillie bound considered by Gromov and Ivanov--Turaev is sharp. We construct a new cocycle representing E and taking only the two values $\pm 2^{-n}$; a null-set obstruction prevents any cocycle from existing on the projective space. We establish the uniqueness of an antisymmetric representative for E in bounded cohomology., Comment: 19 pages

Magnetic properties of CoMnGe thin films of different thicknesses (13, 34, 55, 83, 100 and 200 nm), grown by RF sputtering at 400{\deg}C on single crystal sapphire substrates, were studied using vibrating sample magnetometry (VSM) and conventional or micro-strip line (MS) ferromagnetic resonance (FMR). Their behavior is described assuming a magnetic energy density showing twofold and fourfold in-plane anisotropies with some misalignment between their principal directions. For all the samples, the easy axis of the fourfold anisotropy is parallel to the c-axis of the substrate while the direction of the twofold anisotropy easy axis varies from sample to sample and seems to be strongly influenced by the growth conditions. Its direction is most probably monitored by the slight unavoidable angle of miscut the Al2O3 substrate. The twofold in-plane anisotropy field is almost temperature independent, in contrast with the fourfold field which is a decreasing function of the temperature. Finally, we study the frequency dependence of the observed line-width of the resonant mode and we conclude to a typical Gilbert damping constant of 0.0065 for the 55-nm-thick film., Comment: 7 pages, 7 figures, To be published (Journal of Applied Physics)

We point out a simple characterisation of topological amenability in terms of bounded cohomology, following Johnson's reformulation of amenability.

J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier., Comment: 4 pages

We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterised amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries., Comment: This is the second of a pair of two articles, originally posted on September 2, 2008 as arXiv:0809.0457v1 For the first part, see arXiv:0809.0457v2

Let $\Gamma$ be an irreducible lattice in a product of n infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n is at least 3, then each Kac-Moody groups is in fact a simple algebraic group over a local field and $\Gamma$ is an arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n is at least 2: either $\Gamma$ is an S-arithmetic (hence linear) group, or it is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout., Comment: Subsection 2.B was modified and an example was added there

We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple. Two appendices introduce results and examples around the concept of quasi-product., Comment: Index added; minor changes

We establish a connection between Dixmier's unitarisability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is non-unitarisable if its first L2-Betti number is non-zero or if it is finitely generated with non-trivial cost. Our criterion also applies to torsion groups constructed by D. Osin, thus providing the first examples of non-unitarisable groups not containing a non-Abelian free subgroup.

We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further developments on non-positively curved lattices are exposed in a companion paper: "Isometry groups of non-positively curved spaces: discrete subgroups"., Comment: The original version (September 2, 2008) has been split into two articles. This is the first part; the second is available as of today on the arxiv under the title: "Isometry groups of non-positively curved spaces: discrete subgroups"

We present here high precision magnetisation measurements in polycrystalline $YBa_2Cu_3O_{x}$ samples, with oxygen content ranging from $x=6.19$ to $x=7.00$. By analysing the temperature derivative of the susceptibility, we found in the underdoped superconducting samples a singular point at a temperature corresponding to $T_{mag}$, the temperature below which polarised neutrons experiments have evidenced a symmetry breaking. We believe that this is a thermodynamic indication for the existence of a phase transition in the pseudogap state of underdoped $YBa_2Cu_3O_{x}$., Comment: 12 pages, 6 figures, one bibliographic file

We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity statements are obtained., Comment: 6 pages

In this note not intended for publication, it is observed that a wellnigh trivial application of the ergodic theorem of Karlsson-Ledrappier yields a strong LLN for arbitrary concave moments., Comment: Not for publication. (2 pages)