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.

Musical actions of dihedral groups

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor triads. We illustrate both geometrically and algebraically how these two actions are dual. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

Conjugacy in Thompson's groups

We give a unified solution the conjugacy problem in Thompson’s groups F, V , and T using strand diagrams, and we analyze the complexity of the resulting algorithms.

Solid-phase synthesis and NMR studies of modified oligonucleotides forming triplex helix and of oligonucleopeptides mimicking the topoisomerase I-DNA covalent complex

Report for the scientific sojourn carried out at the Institut de Biologia Molecular de Barcelona of the CSIC –state agency – from april until september 2007. Topoisomerase I is an essential nuclear enzyme that modulates the topological status of DNA, facilitating DNA helix unwinding during replication and transcription. We have prepared the oligonucleotide-peptide conjugate Ac-NLeu-Asn-Tyr(p-3’TTCAGAAGC5’)-LeuC-CONH-(CH2)6-OH as model compound for NMR studies of the Topoisomerase I- DNA complex. Special attention was made on the synthetic aspects for the preparation of this challenging compound especially solid supports and protecting groups. The desired peptide was obtained although we did not achieve the amount of the conjugate needed for NMR studies. Most probably the low yield is due to the intrinsic sensitive to hydrolysis of the phosphate bond between oligonucleotide and tyrosine. We have started the synthesis and the structural characterization of oligonucleotides carrying intercalating compounds. At the present state we have obtained model duplex and quadruplex sequences modified with acridine and NMR studies are underway. In addition to this project we have successfully resolved the structure of a fusion peptide derived from hepatitis C virus envelope synthesized by the group of Dr. Haro and we have synthesized and started the characterization of a modified G-quadruplex.

Mather invariants in groups of piecewise-linear homeomorphisms

We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in [2] and Kassabov and Matucci in [5]. Thanks to the interplay between the techniques, we produce a simplified point of view of conjugacy that allows ua to easily recover centralizers and lends itself to generalization.

The simultaneous conjugacy problem in groups of piecewise linear functions

Guba and Sapir asked, in their joint paper [8], if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give an elementary proof for the solution of the latter question. This relies purely on the description of F as the group of piecewise linear orientation-preserving homeomorphisms of the unit. The techniques we develop allow us also to solve the ordinary conjugacy problem as well, and we can compute roots and centralizers. Moreover, these techniques can be generalized to solve the same questions in larger groups of piecewise-linear homeomorphisms.

On uniform conjugators in torsion-free hyperbolic groups

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Structure theorems for subgroups of homeomorphism groups

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Bounding the residual finiteness of free groups

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Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the raph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.

Higher arithmetic Chow groups

We give a new construction of higher arithmetic Chow groups for quasi-projective arithmetic varieties over a field. Our definition agrees with the higher arithmetic Chow groups defined by Goncharov for projective arithmetic varieties over a field. These groups are the analogue, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. The degree zero group agrees with the arithmetic Chow groups of Burgos. Our new construction is shown to be a contravariant functor and is endowed with a product structure, which is commutative and associative.

Inter-American Court, Crimes Against Humanity and Peacebuilding in South America

The present work contains a general overview of the sentences of the Inter-American Court of Human Rights (IACtHR), which have recognised that crimes against humanity are pre-existing in customary law, and do not prescribe, nor can they be subject to amnesty or pardon. Specific attention is paid to the consequent restrictions and opportunities offered by said verdicts to countries such as Argentina, Chile, Uruguay and Peru, which find themselves in postconflict transition processes and where peace has been negotiated with certain groups and state structures that are responsible for carrying out crimes against humanity. In doing so, special attention is paid to the impact of the recognition of the nature of crimes against humanity on the notion of the principle of legality, stricto sensu; on the development and evolution of the doctrine and the practice of international human rights law in the inter-American context; and finally on the aforementioned processes of transitional justice.

Obtenció de nous anàlegs amb activitat brassinoesteroide mitjançant modelització molecular i síntesi

Els brassinoesteroides són productes naturals que actuen com a potents reguladors del creixement vegetal. Presenten aplicacions prometedores en l’agricultura degut a que, aplicats exògenament, augmenten la qualitat i la quantitat de les collites. Ara bé, el seu ús s’ha vist restringit degut a la seva costosa obtenció. Aquest fet ha motivat la recerca de nous compostos actius més assequibles. En aquest projecte es planteja el disseny i obtenció de nous anàlegs seguint diferents estratègies que impliquen tant l’ús de mètodes de modelització molecular com de síntesi orgànica. La primera d’aquestes estratègies consisteix en buscar compostos actius en bases de dades de compostos comercials a través de processos de Virtual Screening desenvolupats amb mètodes computacionals basats en Camps d’Interacció Molecular. Així, es van establir i interpretar models de Relacions Quantitatives Estructura-Activitat (QSAR) emprant descriptors independents de l’alineament (GRIND) i, amb col•laboració amb la Universitat de Perugia, aquest criteri de cerca es va ampliar amb l’aplicació de descriptors FLAP de nova generació. Una altra estratègia es va basar en intentar substituir l’esquelet esteroide dels brassinoesteroides per una estructura equivalent, fixant com a cadena lateral el grup (R)-hexahidromandelil. S’han aplicat dos criteris: mètodes computacionals basats en models QSAR establerts amb descriptors GRIND i també en la metodologia SHOP (scaffold hopping), i, per altra banda, anàlegs proposats racionalment a partir d’un estudi efectuat sobre disruptors endocrins no esteroïdals. Sobre les estructures trobades s’hi va unir la cadena lateral comercial esmentada per via sintètica, en la qual s’ha hagut de fer un èmfasi especial en grups protectors. En total, 49 estructures es proposen per a ser obtingudes sintèticament. També s’ha treballat en l’obtenció un agonista derivat de l’hipotètic antagonista KM-01. Totes les molècules candidates, ja siguin comercials o obtingudes sintèticament, estant sent avaluades en el test d’inclinació de la làmina d’arròs (RLIT).