Algebraic extensions in free groups

The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical fieldt heoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g. being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations.

The mean dehn functions of Abelian groups

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Traverso's isogeny conjecture for p-divisible groups

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Minimal lenght elements of Thompson's groups F(p)

We describe a method for determining the minimal length of elements in the generalized Thompson's groups F(p). We compute the length of an element by constructing a tree pair diagram for the element, classifying the nodes of the tree and summing associated weights from the pairs of node classifications. We use this method to effectively find minimal length representatives of an element.

Contraction groups in complete Kac-Moody groups

Let G be an abstract Kac-Moody group over a finite field and G the closure of the image of G in the automorphism group of its positive building. We show that if the Dynkin diagram associated to G is irreducible and neither of spherical nor of affine type, then the contraction groups of elements in G which are not topologically periodic are not closed. (In those groups there always exist elements which are not topologically periodic.)

Parabolic and quasiparabolic subgroups of free partially commutative groups

Let Γ be a finite graph and G be the corresponding free partially commutative group. In this paper we study subgroups generated by vertices of the graph Γ, which we call canonical parabolic subgroups. A natural extension of the definition leads to canonical quasiparabolic subgroups. It is shown that the centralisers of subsets of G are the conjugates of canonical quasiparabolic centralisers satisfying certain graph theoretic conditions.

Programa vellesa, activitat física i dependència (Vafid): justificació i bases per a la sistematització

El present estudi es centra en els programes d’activitat física adreçats a persones grans en situació de dependència, que es esenvolupen en grup i que utilitzen el moviment actiu com a principal eina de treball. Es tracta d’una recerca que, per una banda, estudia i analitza les bases teòriques que avalen la importància i la necessitat de l’aplicació d’aquest tipus de programes. Per l’altra, justifica i defineix, amb detall, les línies directrius que han de guiar el seu desenvolupament i la seva aplicació en l’àmbit d’institucions d’atenció a les persones grans (residències, centres de dia i centres sòciosanitaris). El marc conceptual (capítol II) es construeix a partir d’una àmplia recerca bibliogràfica sobre les quatre dimensions d’anàlisi clau (l’envelliment, la dependència, l’atenció a les persones grans i l’activitat física) que fonamenta i justifica la proposta de programa que es fa en la segona part (capítol III). En aquesta es defineixen els referents, les finalitats, els objectius, els recursos que es poden utilitzar, les indicacions bàsiques per a organitzar el treball, els aspectes metodològics essencials per a l’aplicació del programa i les condicions necessàries per a poder-lo implementar. El resultat de tot el procés de recerca i estudi permet dir que, a nivell teòric, l’activitat física és una eina útil, eficaç i amb moltes possibilitats cara a l’atenció a les persones grans en situació de dependència. Que els esmentats programes han de ser fruit d’un procés de planificació, han de considerar les diferents dimensions de l’ésser en interacció constant, han de ser aplicats donant més importància al procés que al producte i que els recursos disponibles s’han d’utilitzar d’acord amb aquests plantejaments. En les línies de futur per a aquesta recerca, es planteja la seva continuació a partir de l’aplicació del programa VAFiD en diferents grups i la seva avaluació (seguint el model d’avaluació responent de Robert Stake) per determinar-ne la coherència i la qualitat.

Discriminating groups: a comprehensive overview

Discriminating groups were introduced by G.Baumslag, A.Myasnikov and V.Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. However they have taken on a life of their own and have been an object of a considerable amount of study. In this paper we survey the large array results concerning the class of discriminating groups that have been developed over the past decade.

The equation xpyq=zr and groups that act freely on Λ-trees

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Orbit decidability and the conjugacy problem for some extensions of groups

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Elements of algebraic geometry and the positive theory of partially commutative groups

The first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theories of the direct factors of G (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group H. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of H has quantifier elimination and that arbitrary first-order formulas lift from H to H * F, where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.

Metric properties of the braided Thompson's groups

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On the intersection of free subgroups in free products of groups

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A classification, up to hyperbolicity, of groups given by 2 generators and one relator of length 8

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Social polarization: introducing distances between and within groups

The measurement of social polarization has received little attention from the literature. The only social polarization index that has been used to measure religious or ethnic polarization (the RQ index) has several shortcomings that are critically discussed in the paper. In particular, that index is not taking into account the existing distance between and within different groups. A couple of axiomatically characterized social polarization indices that overcome these limitations are presented. In the empirical section we show that the rankings of countries according to the levels of polarization change to a great extent when we replace the RQ index by the indices presented in this paper.