Related papers: Infinitary Axiomatizability of Slender and Cotorsi…

Classically, an abelian group $G$ is said to be slender if every homomorphism from the countable product $\mathbb Z^{\mathbb N}$ to $G$ factors through the projection to some finite product $\mathbb Z^n$. Various authors have proposed…

We prove that groups for which every countable subgroup is free ($\aleph_1$-free groups) are n-slender, cm-slender, and lcH-slender. In particular every homomorphism from a completely metrizable group to an $\aleph_1$-free group has an open…

We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable…

An uncountable $\aleph_1$-free group cannot admit a Polish group topology but an uncountable $\aleph_1$-free abelian group can, as witnessed, for example, by the Baer-Specker group $\mathbb{Z}^\omega$; more strongly, $\mathbb{Z}^\omega$ is…

In this paper we show that there exists an uncountable family of finitely generated simple groups with the same positive theory as any non-abelian free group. In particular, these simple groups have infinite $w$-verbal width for all…

We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is $\Sigma^0_3$-complete. However, when we consider groups of infinite rank, it becomes $\Sigma^1_1$-complete,…

We prove that the abstract commensurator of a nonabelian free group, an infinite surface group, or more generally of a group that splits appropriately over a cyclic subgroup, is not finitely generated. This applies in particular to all…

We construct finitely generated torsion-free solvable groups $G$ that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of $G$ are virtually abelian. In particular all finitely generated…

We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation. We show: $\textbf{Theorem}$ (1) If $G$ is a limit model of cardinality…

We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry…

Recently, Paolini and Shelah have constructed absolutely Hopfian torsion-free abelian groups of any given size. In contrast, we show that this is not necessarily the case for absolutely co-Hopfian groups. We use the infinitary logic to show…

We study the link between stably finiteness and stably projectionless-ness for $C^*$-algebras of solvable Lie groups. We show that these two properties are equivalent if the dimension of the group is not divisible by $4$; otherwise, they…

In partial answer to a question posed by Arnie Miller (http://www.math.wisc.edu/~miller/res/problem.pdf) and X. Caicedo, we obtain sufficient conditions for an L_{omega_1,omega} theory to have an independent axiomatization. As a consequence…

Let $\mathcal G$ denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces $\mathcal S\subseteq \mathcal G$ satisfying the following zero-one law: for any sentence $\sigma$ in the…

Examples are given of non-elementary properties that are preserved under C-filtrations for various classes C of Abelian groups. The Baer-Specker group is never the union of a chain of proper subgroups with cotorsionfree quotients.…

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite…

We show that certain algebraic structures lack freeness in the absence of the axiom of choice. These include some subgroups of the Baer-Specker group $\mathbb{Z}^{\omega}$ and the Hawaiian earring group. Applications to slenderness,…

In [9] we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper we show that our construction from [9] satisfies several additional properties of interest. We deduce from this that countable…

We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of…

Relying on the techniques and ideas from our recent paper [13], we prove several anti-classification results for various rigidity conditions in countable abelian and nilpotent groups. We prove three main theorems: (1) the rigid abelian…