Addendum to `Fake Projective Planes'

The addendum updates the results presented in the paper `Fake Projective Plane, Invent Math 168, 321-370 (2007)' and makes some additions and corrections. The fake projective planes are classified into twenty six classes. Together with a recent work of Donald Cartwright and Tim Steger, there is now a complete list of fake projective planes. There are precisely one hundred fake projective planes as complex surfaces classified up to biholomorphism.Comment: A more refined classification is given in the new versio

A lattice in more than two Kac--Moody groups is arithmetic

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 ther

Bound states in N = 4 SYM on T^3: Spin(2n) and the exceptional groups

The low energy spectrum of (3+1)-dimensional N=4 supersymmetric Yang-Mills theory on a spatial three-torus contains a certain number of bound states, characterized by their discrete abelian magnetic and electric 't Hooft fluxes. At weak coupling, the wave-functions of these states are supported near points in the moduli space of flat connections where the unbroken gauge group is semi-simple. The number of such states is related to the number of normalizable bound states at threshold in the supersymmetric matrix quantum mechanics with 16 supercharges based on this unbroken group. Mathematically, the determination of the spectrum relies on the classification of almost commuting triples with semi-simple centralizers. We complete the work begun in a previous paper, by computing the spectrum of bound states in theories based on the even-dimensional spin groups and the exceptional groups. The results satisfy the constraints of S-duality in a rather non-trivial way.Comment: 20 page

Organization of pre-Variscan basement areas at the north-Gondwanan margin

Pre-Variscan basement elements of Central Europe appear in polymetamorphic domains juxtaposed through Variscan and/or Alpine tectonic events. Consequently, nomenclatures and zonations applied to Variscan and Alpine structures, respectively, cannot be valid for pre-Variscan structures. Comparing pre-Variscan relics hidden in the Variscan basement areas of Central Europe, the Alps included, large parallels between the evolution of basement areas of future Avalonia and its former peri-Gondwanan eastern prolongations (e.g. Cadomia, Intra-Alpine Terrane) become evident. Their plate-tectonic evolution from the Late Proterozoic to the Late Ordovician is interpreted as a continuous Gondwana-directed evolution. Cadomian basement, late Cadomian granitoids, late Proterozoic detrital sediments and active margin settings characterize the pre-Cambrian evolution of most of the Gondwana-derived microcontinental pieces. Also the Rheic ocean, separating Avalonia from Gondwana, should have had, at its early stages, a lateral continuation in the former eastern prolongation of peri-Gondwanan microcontinents (e.g. Cadomia, Intra-Alpine Terrane). Subduction of oceanic ridge (Proto-Tethys) triggered the break-off of Avalonia, whereas in the eastern prolongation, the presence of the ridge may have triggered the amalgamation of volcanic arcs and continental ribbons with Gondwana (Ordovician orogenic event). Renewed Gondwana-directed subduction led to the opening of Palaeo-Tethy

Arithmeticity vs. non-linearity for irreducible lattices

We establish an arithmeticity vs. non-linearity alternative for irreducible lattices in suitable product groups, such as for instance products of topologically simple groups. This applies notably to a (large class of) Kac-Moody groups. The alternative relies on a CAT(0) superrigidity theorem, as we follow Margulis' reduction of arithmeticity to superrigidity.Comment: 11 page

The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let \mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=\mathbf{G}(k_v). Let \Gamma be an arithmetic lattice in G and let C=C(\Gamma) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for \Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is \hat{F}_{\omega}, the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example \Gamma=SL_2(\mathcal{O}(S)), where \mathcal{O}(S) is the ring of S-integers in k, with S=\{v\}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of \Gamma on the Bruhat-Tits tree associated with G.Comment: 27 pages, 5 figures, to appear in J. Reine Angew. Mat

Prevalence of Chlamydia abortus in Belgian ruminants

Chlamydia (C.) abortus enzootic abortion still remains the most common cause of reproductive failure in sheep-breeding countries all over the world. Chlamydia abortus in cattle is predominantly associated with genital tract disease and mastitis. In this study, Belgian sheep (n=958), goats (n=48) and cattle (n=1849) were examined, using the ID Screen (TM) Chlamydia abortus indirect multi-species antibody ELISA. In the sheep, the highest prevalence rate was found in Limburg (4.05%). The animals of Antwerp, Brabant and Liege tested negative. The prevalence in the remaining five regions was low (0.24% to 2.74%). Of the nine goat herds, only one herd in Luxembourg was seropositive. In cattle, the highest prevalence rate was found in Walloon Brabant (4.23%). The animals of Limburg and Namur tested negative. The prevalence rate in the remaining seven regions ranged between 0.39% and 4.02%

Overlap properties of geometric expanders

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of the simplices induced by the images of its hyperedges. In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence $\{H_n\}_{n=1}^\infty$ of arbitrarily large $(d+1)$-uniform hypergraphs with bounded degree, for which $\inf_{n\ge 1} c(H_n)>0$. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of $(d+1)$-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant $c=c(d)$. We also show that, for every $d$, the best value of the constant $c=c(d)$ that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete $(d+1)$-uniform hypergraphs with $n$ vertices, as $n\rightarrow\infty$. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any $d$ and any $\epsilon>0$, there exists $K=K(\epsilon,d)\ge d+1$ satisfying the following condition. For any $k\ge K$, for any point $q \in \mathbb{R}^d$ and for any finite Borel measure $\mu$ on $\mathbb{R}^d$ with respect to which every hyperplane has measure $0$, there is a partition $\mathbb{R}^d=A_1 \cup \ldots \cup A_{k}$ into $k$ measurable parts of equal measure such that all but at most an $\epsilon$-fraction of the $(d+1)$-tuples $A_{i_1},\ldots,A_{i_{d+1}}$ have the property that either all simplices with one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain $q$