Filling in solvable groups and in lattices in semisimple groups

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all Q-rank one lattices in semisimple groups of R-rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion constant.Comment: 7 figure

Compression functions of uniform embeddings of groups into Hilbert and Banach spaces

We construct finitely generated groups with arbitrary prescribed Hilbert space compression \alpha from the interval [0,1]. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the E-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 3, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (respectively, exact, of finite asymptotic dimension) with Hilbert space compression 0 are given. These groups are also the first examples of groups with uniformly convex Banach space compression 0.Comment: 21 pages; version 3: The final version, accepted by Crelle; version 2: corrected misprints, added references, the group has asdim at most 2, not at most 3 as in the first version (thanks to A. Dranishnikov); version 3: took into account referee remarks, added references. the paper is accepted in Crell

Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams

Let Gamma be a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. We construct Makanin-Razborov diagrams for Gamma. We also prove that every system of equations over Gamma is equivalent to a finite subsystem, and a number of structural results about Gamma-limit groups.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper54.abs.htm

Compression functions of uniform embeddings of groups into Hilbert and Banach spaces

We construct finitely generated groups with arbitrary prescribed Hilbert space compression α ∈ [0, 1]. This answers a question of E. Guentner and G. Niblo. For a large class of Banach spaces ℰ (including all uniformly convex Banach spaces), the ℰ-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 2, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (moreover, of finite asymptotic dimension and exact) with Hilbert space compression 0 are given. These groups are also the first examples of groups with uniformly convex Banach space compression

Dynamic characterizations of quasi-isometry and applications to cohomology

We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we give conceptual explanations for previous results of Shalom and Sauer on coarse invariance of homological and cohomological dimensions and Shalom's property $H_{FD}$. As another application, we show that group homology and cohomology in a class of coefficients, including all induced and co-induced modules, are coarse invariants. We deduce that being of type $FP_n$ (over arbitrary rings) is a coarse invariant, and that being a (Poincar\'e) duality group over a ring is a coarse invariant among all groups which have finite cohomological dimension over that ring. Our results also imply that every self coarse embedding of a Poincar\'e duality group over an arbitrary ring must be a coarse equivalence.Comment: 29 pages; improved results and exposition; added and updated reference

Limit groups for relatively hyperbolic groups, I: The basic tools

We begin the investigation of Gamma-limit groups, where Gamma is a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of Drutu and Sapir, we adapt the results from math.GR/0404440 to this context. Specifically, given a finitely generated group G, and a sequence of pairwise non-conjugate homomorphisms {h_n : G -> Gamma}, we extract an R-tree with a nontrivial isometric G-action. We then prove an analogue of Sela's shortening argument.Comment: 41 pages. The new version of this paper has been substantially rewritten. It now includes all of the results of the previous version, and also of math.GR/0408080. The exception to this is the proof of the Hopf property, which follows imediately from Theorem 5.2 of math.GR/0503045 (and does not use anything omitted from this version

Khinchin theorem for integral points on quadratic varieties

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.Comment: 19 page

Researches regarding the influence of distance between rows on the stem and fiber yields at some monoecious hemp varieties under the center of Moldavia pedoclimatic conditions

Hemp is part of the textile plants group with high-value for human and industrial use. Hemp has over 25,000 uses, ranging from food, paints and fuels to clothing and building materials. Hemp is currently considered to be a plant of increasing importance for Europe, being used for fiber and oil extraction and as medicinal plant. Hemp fibers are the most resistant plant fibers and as such, in the past, they were the most prized raw material of the textile industry worldwide.In this paper we present the results regarding the evolution of monoecious hemp crop on the production of stems and fiber, under the pedoclimatic conditions of the Center of Moldova, between 2012 – 2015. The biological material used was represented by three varieties created at A.R.D.S. Secuieni, respectively, Denise, Diana and Dacia and were sown at a distance of 12.5 cm, 25 cm respectively 50 cm between rows. The obtained results revealed that the studied factors influenced to a great extent the production of stems, which varied widely, ranging from 8113 kg / ha to the Denise variety at a distance of 12.5 cm in 2015 (agricultural year characterized as very dry from pluviometric point of view), and the highest yields were obtained at the Denise variety of 15683 kg / ha, at a distance of 25 cm in 2013 (agricultural year characterized as normal from rainfall point of view). On average, for the four years studied, the highest obtained production of fiber was achieved by Dacia variety, at 12.5 cm, of 3388 kg / ha, and the lowest yield of 2546 kg / ha was achieved by Denise variety at a distance of 50 cm between row

The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type $HF^\infty$, i.e. that has a classifying space with the homotopy type of a cellular complex with finitely many cells in each dimension, we show that the isocohomological property is equivalent to the universal cover of the classifying space satisfying polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable (respectively $HF^\infty$ and isocohomological), then we show that the group itself has these respective properties too. Combining with the results of Connes-Moscovici and Dru{\c{t}}u-Sapir we conclude that a group satisfies the Novikov conjecture if it is relatively hyperbolic to subgroups that are of property RD, of type $HF^\infty$ and isocohomological.Comment: 35 pages, no figure