Each second countable abelian group is a subgroup of a second countable divisible group

Minimally almost periodic group topology on countable torsion Abelian groups

For any countable torsion subgroup $H$ of an unbounded Abelian group $G$ there is a complete Hausdorff group topology $τ$ such that $H$ is the von Neumann radical of $(G,τ)$. In particular, any unbounded torsion countable Abelian group admits a complete Hausdorff minimally almost periodic (MinAP) group topology. If $G$ is a bounded torsion countably infinite Abelian group, then it admits a MinAP group topology if and only if all its leading Ulm-Kaplansky invariants are infinite. In such a case, a MinAP group topology can be chosen to be complete.

Minimally almost periodic group topology on infinite countable Abelian groups: A solution to Comfort's question

For any countable subgroup $H$ of an unbounded Abelian group $G$ there is a complete Hausdorff group topology $τ$ such that $H$ is the von Neumann radical of $(G,τ)$. In particular, we obtain the positive answer to Comfort's question: any unbounded countable Abelian group admits a complete Hausdorff minimally almost periodic (MinAP) group topology. A bounded infinite Abelian group admits a MinAP group topology if and only if all its leading Ulm-Kaplansky invariants are infinite. If, in addition, $G$ is countably infinite, a MinAP group topology can be chosen to be complete.

On large sequential groups

We construct, using $\diamondsuit$, an example of a sequential group $G$ such that the only countable sequential subgroups of $G$ are closed and discrete, and the only quotients of $G$ that have a countable pseudocharacter are countable and Fréchet. We also show how to construct such a $G$ with several additional properties (such as make $G^2$ sequential, and arrange for every sequential subgroup of $G$ to be closed and contain a nonmetrizable compact subspace, etc.). Several results about $k_ω$ sequential groups are proved. In particular, we show that each such group is either locally compact…

Locally Minimal Topological Groups 2

We continue in this paper the study of locally minimal groups started in \cite{LocMin}. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian groups containing dense countable locally minimal subgroups, as well as those containing dense locally minimal subgroups of countable free-rank. We also characterize the compact abelian groups whose torsion part is dense and locally minimal. We call a topological group $G$ {\it almost minimal} if it has a closed, minimal normal subgroup $N$ such that the qu…

Cardinal invariants and convergence properties of locally minimal groups

If G is a locally essential subgroup of a compact abelian group K, then: (i) t(G)=w(G)=w(K), where t(G) is the tightness of G; (ii) if G is radial, then K must be metrizable; (iii) G contains a super-sequence S converging to 0 such that |S|=w(G)=w(K). Items (i)--(iii) hold when G is a dense locally minimal subgroup of K. We show that locally minimal, locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007. For every uncountable cardinal kappa, we cons…

Non metrizable topologies on Z with countable dual group

In this paper we give two families of non-metrizable topologies on the group of the integers having a countable dual group which is isomorphic to a infinite torsion subgroup of the unit circle in the complex plane. Both families are related to $D$-sequences, which are sequences of natural numbers such that each term divides the following. The first family consists of locally quasi-convex group topologies. The second consists of complete topologies which are not locally quasi-convex. In order to study the dual groups for both families we need to make numerical considerations of independent inte…

可数稠密齐性与拓扑群

Countable dense homogeneity and topological groups

本研究基于Medvedev成果构造出ZFC下大小为b的非Polish可数稠密齐性拓扑群示例（为ℤ^ω的λ-集稠密子群），还提出无孤立点的该类Baire群含Cantor集拷贝的猜想并在特殊情形下得证。

$C^*$-单群的等变随机子群

Invariant random subgroups of $C^*$-simple groups

本文证明可数离散 $C^*$-单群的等变随机子群（IRS）几乎必然仍是 $C^*$-单群，推广了单群性质到随机子群语境。

Countable approximation of topological $G$-manifolds, III: arbitrary Lie groups $G$

The Hilbert-Smith conjecture states, for any connected topological manifold $M$, any locally compact subgroup of $\mathrm{Homeo}(M)$ is a Lie group. We generalize basic results of Segal-Kosniowski-tomDieck (2.6), James-Segal (2.12), G Bredon (3.7), Jaworowski-Antonyan et al. (5.5), and E Elfving (7.3). The last is our main result: for any Lie group $G$, any Palais-proper topological $G$-manifold has the equivariant homotopy type of a countable proper $G$-CW complex. Along the way, we verify an $n$-classifying space for principal $G$-bundles (5.10).

Sequences and filters of characters characterizing subgroups of compact abelian groups

Let H be a countable subgroup of the metrizable compact abelian group G and f:H -> T=R/Z a (not necessarily continuous) character of H. Then there exists a sequence (chi_n)_n of (continuous) characters of G such that lim_n chi_n(alpha) = f(alpha) for all alpha in H and (chi_n(alpha))_n does not converge whenever alpha in G\H. If one drops the countability and metrizability requirement one can obtain similar results by using filters of characters instead of sequences. Furthermore the introduced methods allow to answer questions of Dikranjan et al.

可数结构的极可 amenable 自动同构群

Extremely amenable automorphism groups of countable structures

研究极可 amenable 群的数量及其作为可数结构自动同构群的构造，证明存在连续统多个非同构的极可 amenable 群。

On the Borel Complexity of Characterized Subgroups

In a compact abelian group $X$, a characterized subgroup is a subgroup $H$ such that there exists a sequence of characters $\vs=(v_n)$ of $X$ such that $H=\{x\in X:v_n(x)\to 0 \text{ in } \T\}$. Gabriyelyan proved for $X=\T$, that $\{x\in\T:n!x\to 0 \text{ in }\T\}$ is not an $F_σ$-set. In this paper, we give a complete description of the $F_σ$-subgroups of $\T$ characterized by sequences of integers $\vs=(v_n)$ such that $v_n|v_{n+1}$ for all $n\in\N$ (we show that these are exactly the countable characterized subgroups). Moreover in the general setting of compact metrizable abelian groups, w…