On abstract commensurators of groups

We prove that the abstract commensurator of a nonabelian free group, an infinite surface group, or more generally of a group that splits appropriately over a cyclic subgroup, is not finitely generated. This applies in particular to all torsion-free word-hyperbolic groups with infinite outer automorphism group and abelianization of rank at least 2. We also construct a finitely generated, torsion-free group which can be mapped onto Z and which has a finitely generated commensurator.Comment: 13 pages, no figur

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

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

The inflammatory response of primary bovine mammary epithelial cells to Staphylococcus aureus strains is linked to the bacterial phenotype

Staphylococcus aureus is a major mastitis-causing pathogen in dairy cows. The latex agglutination-based Staphaurex test allows bovine S. aureus strains to be grouped into Staphaurex latex agglutination test (SLAT)-negative [SLAT(2)] and SLATpositive [SLAT(+)] isolates. Virulence and resistance gene profiles within SLAT(2) isolates are highly similar, but differ largely from those of SLAT(+) isolates. Notably, specific genetic changes in important virulence factors were detected in SLAT(2) isolates. Based on the molecular data, it is assumed that SLAT(+) strains are more virulent than SLAT(2) strains. The objective of this study was to investigate if SLAT(2) and SLAT(+) strains can differentially induce an immune response with regard to their adhesive capacity to epithelial cells in the mammary gland and in turn, could play a role in the course of mastitis. Primary bovine mammary epithelial cells (bMEC) were challenged with suspensions of heat inactivated SLAT(+) (n = 3) and SLAT(2) (n = 3) strains isolated from clinical bovine mastitis cases. After 1, 6, and 24 h, cells were harvested and mRNA expression of inflammatory mediators (TNF-a, IL-1b, IL-8, RANTES, SAA, lactoferrin, GM-CSF, COX-2, and TLR-2) was evaluated by reverse transcription and quantitative PCR. Transcription (DDCT) of most measured factors was induced in challenged bMEC for 6 and 24 h. Interestingly, relative mRNA levels were higher (P,0.05) in response to SLAT(+) compared to SLAT(2) strains. In addition, adhesion assays on bMEC also showed significant differences between SLAT(+) and SLAT(2) strains. The present study clearly shows that these two S. aureus strain types cause a differential immune response of bMEC and exhibit differences in their adhesion capacity in vitro. This could reflect differences in the severity of mastitis that the different strain types may induce

Automorphism groups of polycyclic-by-finite groups and arithmetic groups

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(\Gamma, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory

Growth in solvable subgroups of GL_r(Z/pZ)

Let $K=Z/pZ$ and let $A$ be a subset of \GL_r(K) such that is solvable. We reduce the study of the growth of $A$ under the group operation to the nilpotent setting. Specifically we prove that either $A$ grows rapidly (meaning $|A\cdot A\cdot A|\gg |A|^{1+\delta}$), or else there are groups $U_R$ and $S$, with $S/U_R$ nilpotent such that $A_k\cap S$ is large and $U_R\subseteq A_k$, where $k$ is a bounded integer and $A_k = \{x_1 x_2...b x_k : x_i \in A \cup A^{-1} \cup {1}}$. The implied constants depend only on the rank $r$ of $\GL_r(K)$. When combined with recent work by Pyber and Szab\'o, the main result of this paper implies that it is possible to draw the same conclusions without supposing that is solvable.Comment: 46 pages. This version includes revisions recommended by an anonymous referee including, in particular, the statement of a new theorem, Theorem

Syzygies in equivariant cohomology for non-abelian Lie groups

We extend the work of Allday-Franz-Puppe on syzygies in equivariant cohomology from tori to arbitrary compact connected Lie groups G. In particular, we show that for a compact orientable G-manifold X the analogue of the Chang-Skjelbred sequence is exact if and only if the equivariant cohomology of X is reflexive, if and only if the equivariant Poincare pairing for X is perfect. Along the way we establish that the equivariant cohomology modules arising from the orbit filtration of X are Cohen-Macaulay. We allow singular spaces and introduce a Cartan model for their equivariant cohomology. We also develop a criterion for the finiteness of the number of infinitesimal orbit types of a G-manifold.Comment: 28 pages; minor change

Chaos for Liouville probability densities

Using the method of symbolic dynamics, we show that a large class of classical chaotic maps exhibit exponential hypersensitivity to perturbation, i.e., a rapid increase with time of the information needed to describe the perturbed time evolution of the Liouville density, the information attaining values that are exponentially larger than the entropy increase that results from averaging over the perturbation. The exponential rate of growth of the ratio of information to entropy is given by the Kolmogorov-Sinai entropy of the map. These findings generalize and extend results obtained for the baker's map [R. Schack and C. M. Caves, Phys. Rev. Lett. 69, 3413 (1992)].Comment: 26 pages in REVTEX, no figures, submitted to Phys. Rev.

Quasi-analyticity and determinacy of the full moment problem from finite to infinite dimensions

This paper is aimed to show the essential role played by the theory of quasi-analytic functions in the study of the determinacy of the moment problem on finite and infinite-dimensional spaces. In particular, the quasi-analytic criterion of self-adjointness of operators and their commutativity are crucial to establish whether or not a measure is uniquely determined by its moments. Our main goal is to point out that this is a common feature of the determinacy question in both the finite and the infinite-dimensional moment problem, by reviewing some of the most known determinacy results from this perspective. We also collect some properties of independent interest concerning the characterization of quasi-analytic classes associated to log-convex sequences.Comment: 28 pages, Stochastic and Infinite Dimensional Analysis, Chapter 9, Trends in Mathematics, Birkh\"auser Basel, 201

Закономерности распределения аномальных концентраций гелия в палеозойских отложениях Донбасса

Підвищені концентрації гелію у вільних і розчинених газах у вугільно-породному масиві Донбасу приурочені до зон зчленування Донбасу з Приазовським і Воронезьким кристалічними масивами і Головної антикліналі. Диз'юнктивні тектонічні порушення глибокого закладення служили, ймовірно, шляхами транзиту гелію від кристалічних порід фундаменту в палеозойські відклади Донбасу, і підвищений вміст гелію може служити індикатором шляхів транзиту газів з глибоких горизонтів Донецького вугільного басейну.Elevated concentrations of helium in free and dissolved gases in coal-rock mass of the Donets Coal Basin are confined to the junction zones of the Donets Coal Basin with the Priazovie and Voronezh crystalline core-areas and Glavnaya (Principal) Anticline. Deep-laid disjunctive dislocations were probably the ways for transit of helium from crystalline basement rocks into Paleozoic deposits of the Basin. Elevated concentrations of helium can also serve as indicators of the ways for transit of gases from deep levels of the Donets Coal Basin