Cellular covers of local groups

We prove that, in the category of groups, the composition of a cellularization and a localization functor need not be idempotent. This provides a negative answer to a question of Emmanuel Dror Farjoun.Ministerio de Educación y CienciaJunta de Andalucí

On the idempotency of some composite functors

We present examples of localization functors whose composition with certain cellularization functors is not idempotent, and vice versa.Ministerio de Educación y Cienci

Cryptography with right-angled Artin groups

In this paper we propose right-angled Artin groups as a platform for secret sharingschemes based on the efficiency (linear time) of the word problem. Inspired by previous work of Grigoriev-Shpilrain in the context of graphs, we de ne two new problems: SubgroupIsomorphism Problem and Group Homomorphism Problem. Based on them, we also proposetwo new authentication schemes. For right-angled Artin groups, the Group Homomorphismand Graph Homomorphism problems are equivalent, and the later is known to be NP-complete.In the case of the Subgroup Isomorphism problem, we bring some results due to Bridson whoshows there are right-angled Artin groups in which this problem is unsolvable.Professional Staff Congress-City University of New YorkCity Tech FoundationOffice of Naval ResearchEuropean Research CouncilNational Science FoundationMinisterio de Ciencia e Innovació

Cryptography with right-angled Artin groups

In this paper we propose right-angled Artin groups as a platform for secret sharingschemes based on the efficiency (linear time) of the word problem. Inspired by previous work of Grigoriev-Shpilrain in the context of graphs, we de ne two new problems: SubgroupIsomorphism Problem and Group Homomorphism Problem. Based on them, we also proposetwo new authentication schemes. For right-angled Artin groups, the Group Homomorphismand Graph Homomorphism problems are equivalent, and the later is known to be NP-complete.In the case of the Subgroup Isomorphism problem, we bring some results due to Bridson whoshows there are right-angled Artin groups in which this problem is unsolvable.Professional Staff Congress-City University of New YorkCity Tech FoundationOffice of Naval ResearchEuropean Research CouncilNational Science FoundationMinisterio de Ciencia e Innovació

Torsion homology and cellular approximation

We describe the role of the Schur multiplier in the structure of the p-torsion of discrete groups. More concretely, we show how the knowledge of H2G allows us to approximate many groups by colimits of copies of p-groups. Our examples include interesting families of noncommutative infinite groups, including Burnside groups, certain solvable groups and branch groups. We also provide a counterexample for a conjecture of Emmanuel Farjoun.Fondo Europeo de Desarrollo RegionalMinisterio de Ciencia e InnovaciónConsejería de Economía, Innovación y Ciencia (Junta de Andalucía

The cellular structure of the classifying spaces of finite groups

In this paper we obtain a description of the BZ/p-cellularization (in the sense of Dror-Farjoun) of the classifying spaces of all finite groups, for all primes p.Ministerio de Educación y Cienci

On the classifying space for proper actions of groups with cyclic torsion

In this paper we introduce a common framework for describing the topological part of the Baum-Connes conjecture for a wide class of groups. We compute the Bredon homology for groups with aspherical presentation, one-relator quotients of products of locally indicable groups, extensions of Zn by cyclic groups, and fuchsian groups. We take advantage of the torsion structure of these groups to use appropriate models of the universal space for proper actions which allow us, in turn, to extend some technology defined by Mislin in the case of one-relator groups.Ministerio de Ciencia e InnovaciónEngineering and Physical Sciences Research Counci

Homotopy idempotent functors on classifying spaces

Fix a prime p. Since their definition in the context of Localization Theory, the homotopy functors PBZ/p and CWBZ/p have shown to be powerful tools to understand and describe the mod p structure of a space. In this paper, we study the effect of these functors on a wide class of spaces which includes classifying spaces of compact Lie groups and their homotopical analogues. Moreover, we investigate their relationship in this context with other relevant functors in the analysis of the mod p homotopy, such as Bousfield-Kan completion and Bousfield homological localization

Localización, acciones propias y espacios clasificadores de grupos discretos

Consultable des del TDXTítol obtingut de la portada digitalitzadaLa principal aportación de este trabajo ha sido encontrar un funtor de localización que pasa de modo natural de modelos del espacio clasificador clásico de un grupo discreto G a modelos del espacio clasificador para G-fibrados propios. Más concretamente, hemos obtenido lo siguiente: Teorema. Si G es un grupo discreto cuya dimensión geométrica propia es finita y P es el funtor de anulación con respecto al wedge de todos los espacios clasificadores de grupos cíclicos de orden primo, se tiene que PBG posee el tipo de homotopía del espacio clasificador para G-fibrados propios B_G. La prueba tiene esencialmente dos ingredientes: un modelo particular para E_G como construcción de Grothendieck de un funtor sobre una categoría cuyo nervio es un modelo del espacio de órbitas B_G, y la solución de Miller a la conjetura de Sullivan. El teorema que acabamos de citar ha sido utilizado de tres maneras diferentes en la tesis: primero, lo hemos usado directamente para obtener información sobre la estructura homotópica de B_G; segundo, lo hemos aplicado para traducir propiedades homotópicas de BG y obtener modelos concretos de B_G vía funtores de localización; y tercero, modelos geométricos bien conocidos de B_G nos han permitido calcular la BZ/p-anulación de los espacios clasificadores de algunas familias de grupos discretos. Nuestro interés en localizaciones de BG para G finito fue originalmente como paso intermedio en la demostración del teorema que acabamos de citar. Si embargo, estas cuestiones pronto adquirieron interés independiente, de modo que procedimos a un estudio más detallado que en particular incluyó a la celularización (que en cierto modo es dual de la anulación). El principal resultado obtenido en este contexto fue el siguiente: Proposición. Si p es un primo, G un grupo finito y T el subgrupo normal minimal de G que contiene a toda la p-torsión, la BZ/p-anulación PBG está caracterizada por una fibración Y ‡ PBG ‡ B(G/T), donde Y denota al producto de las q-compleciones de BT para todos los primos diferentes de p. La demostración se lleva a cabo utilizando técnicas de teoría de homotopía, como cuadrados aritméticos, descomposición homológicas o preservación de fibraciones bajo funtores de localización. El principal resultado sobre BZ/p-celularización es la clasificación de los grupos finitos tales que su espacio clasificador es BZ/p-celular: Proposición. Si p es un primo y G es un grupo finito, el espacio clasificador de G es BZ/p-celular si y sólo si G es un p-grupo generado por elementos de orden p. Nuestro estudio de la BZ/p-celularización de BG es realizado comparando esta construcción con la Z/p-celularización de grupos estudiada a finales de lo noventa por varios autores, y los resultados obtenidos pueden considerarse la extensión de resultados ya conocidos sobre espacios de Moore M(Z/p,1) al caso infinito-dimensional.The main achievement of our work has been to find a localization functor that passes in a natural way from models of the classical classifying space of a discrete group G to its classifying space for proper G-bundles. More concretely, we have obtained the following: Theorem. If G is a discrete group whose proper geometrical dimension is finite and P is the nullification with regard to the wedge of the classifying spaces of all primes, then PBG has the homotopy type of the classifying space for proper G-bundles B_G. The proof has essentially two ingredients: a particular model for E_G as the Grothendieck construction of a functor over a category whose nerve is a model for the orbit space B_G, and the Miller solution to the Sullivan conjecture. We have used the main theorem in three ways: first, we have used it directly to find information about the homotopy structure of B_G; second, we have applied it to translate homotopical properties of BG to obtain concrete models of B_G via localization functors; and last, well-known geometrical models of B_G have allowed us the computation of the BZ/p-nullification of the classifying spaces of some families of discrete groups. Our interest on localizations of BG for G finite was originally as an intermediate step in the proof of the theorem we have just quoted. However, these questions soon acquired independent interest, so we made a detailed study, including the cellularization, that is in some dual of the nullification. The main result in this context have been the following: Proposition. If p is a prime, G is a finite group and T is the minimal normal subgroup of G that contains all the p-torsion, the BZ/p-nullification PBG is characterized by a fibration Y ‡ PBG ‡ B(G/T), where Y stands for the product of q-completions of BT for all the primes different of p. The proof is carried out using homotopy techniques, just like arithmetic squares, homological decompositions or preservation of fibrations under localization functors. The main result about BZ/p-cellularization is a classification of BZ/p-cellular classifying spaces of finite groups: Proposition. If p is a prime and G is a finite group, the classifying space of G is BZ/p-cellular if and only if G is a p-group generated by order p elements. We study the BZ/p-cellularization of BG by comparing this construction with the Z/p-cellularization of groups studied in the last nineties by several authors, and our research on this topic can be considered as an extension of work already done on Moore spaces M(Z/p,1) to the infinite-dimensional case

The Shapley group value

Following the original interpretation of the Shapley value (Shapley, 1953a) as a priori evaluation of the prospects of a player in a multi-person iteraction situation, we propose a group value, which we call the Shapley group value, as a priori evaluation of the prospects of a group of players in a coalitional game when acting as a unit. We study its properties and we give an axiomatic characterization. We motivate our proposal by means of some relevant applications of the Shapley group value, when it is used as an objective function by a decision maker who is trying to identify an optimal group of agents in a framework in which agents interact and the attained benefit can be modeled by means of a transferable utility game. As an illustrative example we analyze the problem of identifying the set of key agents in a terrorist network.This research has been supported by I+D+i research project MTM2011-27892 from the Government of Spai