Related papers: Topologically independent sets in precompact group…

We prove that no quantifier-free formula in the language of group theory can define the $\aleph_1$-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of…

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless $G$ is isomorphic to a free product of free and surface groups, every finite abelian group $M$ appears as a direct summand in…

$\aleph_1$-free groups, abelian groups for which every countable subgroup is free, exhibit a number of interesting algebraic and set-theoretic properties. In this paper, we give a complete proof that the property of being $\aleph_1$-free is…

We adapt Safin's result on powers of sets in free groups to obtain Helfgott type growth in free products: if A is any finite subset of a free product of two arbitrary groups then either A is conjugate into one of the factors, or the size of…

This article is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic…

We prove that a large class of metrizable group topologies for subgroups of $\mathbb{R}^n$ and the completions of the subgroups are locally isometric to, respectively, metrizable group topologies for $\mathbb{Z}$ and their completions,…

The existence of a countably compact group without non-trivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boolean topological group G without non-trivial convergent…

A maximal abelian normal subgroup A in a nilpotent group N is self-centralizing. This makes their role an important one in determining the structure of the nilpotent group. For example if A is finite then N is also finite. In the free…

We give an introduction to the Cayley-Abels graph for a totally disconnected, locally compact (tdlc) group. It is a generalization of the Cayley graph. We illustrate that on the one hand, Cayley-Abels graphs are useful tools to extend…

We study the topological structure of the automorphism groups of compact quantum groups showing that, in parallel to a classical result due to Iwasawa, the connected component of identity of the automorphism group and of the "inner"…

We investigate topologies on groups which arise naturally from their algebraic structure, including the Frech\'et-Markov, Hausdorff-Markov, and various kinds of Zariski topologies. Answering a question by Dikranjan and Toller, we show that…

Let $T$ be a theory which is t-minimal, meaning that with respect to some definable topology, a unary definable set $D \subseteq M$ has non-empty interior iff it is infinite. If $K$ is a definable field in $T$, then $K$ is finite or "large"…

In this paper, we pose the concepts of pre-topological groups and some generalizations of pre-topological groups. First, we systematically investigate some basic properties of pre-topological groups; in particular, we prove that each…

Let $\tau$ be an uncountable cardinal. The notion of a \emph{$\tau$-fine} topological group was introduced in 2021. More recently, H. Zhang et al. generalized this concept by defining pseudo-$\tau$-fine topological groups to study certain…

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…

We prove that the group G=Hom(P,Z) of all homomorphisms from the Baer-Specker group P to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that…

The topological group version of the celebrated Banach-Mazur problem asks wether every infinite topological group has a non-trivial separable quotient group. It is known that compact groups have infinite separable metrizable quotient…

For a finite multiset $A$ of an abelian group $G$, let $\text{FS}(A)$ denote the multiset of the $2^{|A|}$ subset sums of $A$. It is natural to ask to what extent $A$ can be reconstructed from $\text{FS}(A)$. We fully solve this problem for…

Topological groups whose underlying spaces are basically disconnected, $F$-, or $F'$-spaces but not $P$-spaces are considered. It is proved, in particular, that the existence of a Lindel\"of basically disconnected topological group which is…

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of…