On localizations of quasi-simple groups with given countable center

A group homomorphism $i: H \to G$ is a localization of $H$ if for every homomorphism $\varphi: H\rightarrow G$ there exists a unique endomorphism $\psi: G\rightarrow G$, such that $i \psi=\varphi$ (maps are acting on the right). G\"{o}bel and Trlifaj asked in \cite[Problem 30.4(4), p. 831]{GT12} which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e. a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Th\'{e}venaz and Viruel.Comment: 21 page

Generators and closed classes of groups

We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. We also establish several new relations between singly-generated closed classes.Comment: 22 page

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

Expanders and right-angled Artin groups

The purpose of this article is to give a characterization of families of expander graphs via right-angled Artin groups. We prove that a sequence of simplicial graphs $\{\Gamma_i\}_{i\in\mathbb{N}}$ forms a family of expander graphs if and only if a certain natural mini-max invariant arising from the cup product in the cohomology rings of the groups $\{A(\Gamma_i)\}_{i\in\mathbb{N}}$ agrees with the Cheeger constant of the sequence of graphs, thus allowing us to characterize expander graphs via cohomology. This result is proved in the more general framework of \emph{vector space expanders}, a novel structure consisting of sequences of vector spaces equipped with vector-space-valued bilinear pairings which satisfy a certain mini-max condition. These objects can be considered to be analogues of expander graphs in the realm of linear algebra, with a dictionary being given by the cup product in cohomology, and in this context represent a different approach to expanders that those developed by Lubotzky-Zelmanov and Bourgain-Yehudayoff.Comment: 21 pages. Accepted version. To appear in J. Topol. Ana