Related papers: Universal subgroups of Polish groups

Let $\mathfrak F$ be a formation and let $G$ be a group. A subgroup $H$ of $G$ is $\mathrm{K}\mathfrak F$-subnormal (submodular) in $G$ if there is a subgroup chain $H=H_0\le \ H_1 \le \ \ldots \le H_i \leq H_{i+1}\le \ldots \le \ H_n=G$…

Given a $T_0$ paratopological group $G$ and a class $\mathcal C$ of continuous homomorphisms of paratopological groups, we define the $\mathcal C$-$semicompletion$ $\mathcal C[G)$ and $\mathcal C$-$completion$ $\mathcal C[G]$ of the group…

Throughout this Abstract, $G$ is a topological Abelian group and $\hat{G}$ is the space of continuous homomorphisms from $G$ into $T$ in the compact-open topology. A dense subgroup $D$ of $G$ determines $G$ if the (necessarily continuous)…

Let $G$ be a connected and simply connected semisimple algebraic group over $\Bbb Q$ and let $\Gamma\subset G(\Bbb Q)$ be an arithmetic subgroup. Let $K_\infty\subset G(\Bbb R)$ be a maximal compact subgroup and let $d$ be the dimension of…

For an abelian topological group G let G^* denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G) and an open…

For a compactly generated LCA group $G$, it is shown that the set $H(G)$ of all generalized characters on $G$ equipped with the compact-open topology is a LCA group and $H(G) = \dg$ (the dual group of $G$) if and only if $G$ is compact.…

We give a proof of the existence of generalized definable locally compact models for arbitrary approximate subgroups via an application of topological dynamics in model theory. Our construction is simpler and shorter than the original one…

Let $X$ be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group $G$ on $X.$ If $G$ and $X$ are Polish spaces, we show…

We associate a rigid C*-tensor category $C$ to a totally disconnected locally compact group $G$ and a compact open subgroup $K < G$. We characterize when $C$ has the Haagerup property or property (T), and when $C$ is weakly amenable. When…

Let G be a finite group and {\sigma} = {{\sigma}_i, i \in I} be a partition of the set of all primes \mathbb{P}. A set \mathcal{H} of subgroups of G with 1 \in \mathcal{H} is said to be a complete Hall {\sigma}-set of G if every…

Stanislaw Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). J\'anos Pach (1981) answered this question in the negative. We…

For every countable group G we construct a compact path connected subspace K of R^4 whose fundamental group is isomorphic to G. Our construction is much simpler than the one found recently by Virk.

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well known examples from…

Let G be a locally compact topological group and X a compact space with continuous G-action. The main result of this essay states that the following statements are equivalent : 1) The action of G on X is topologically amenable ; 2) Every…

Hyperspaces $\mathcal H(X)$ of all countable compact subsets of a metric space $X$ and $\mathcal A_n(X)$ of infinite compact subsets which have at most $n$ ($n\in\mathbb N$), or finitely many ($n=\omega$) or countably many ($n=\omega+1$)…

In 'The essentially chief series of a compactly generated locally compact group', an analogue of chief series for finite groups is discovered for compactly generated locally compact groups. In the present article, we show that chief factors…

We prove that a homomorphism $h:X\to Y$ from a (locally compact) Cech-complete topological group $X$ to a topological group $Y$ is continuous if and only if $h$ is Borel-measurable if and only if $h$ is universally measurable (if and only…

We prove definable versions of the Universal Coefficient Theorems of Eilenberg--Mac Lane expressing the (Steenrod) homology groups of a compact metrizable space in terms of its integral cohomology groups, and the (\v{C}ech) cohomology…

We introduce the concept of an $\mathrm{L}^{1}$ full group associated with a measure-preserving action of a Polish normed group on a standard probability space. These groups carry a natural Polish group topology induced by an $\mathrm{L}^1$…

We embed a countably categorical group G into a locally compact group c(G) with a non-trivial topology and study how topological properties of c(G) are connected with the structure of definable subgroups of G.