Compact-like abelian groups without non-trivial quasi-convex null sequences

零伦等变移位与Borsuk-Ulam相容紧致阿贝尔群

Nullhomotopic equivariant shifts and Borsuk-Ulam-compatible compact abelian groups

本文否证了Borsuk-Ulam风格猜想，指出其仅在无挠profinite紧致阿贝尔群情形下失效，并给出精确刻画。

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that for every locally compact abelian group G, the following statements are equivalent: (i) G contains no sequence {x_n} such that {0} \cup {\pm x_n : n \in N} is infinite and quasi-convex in G, and x_n --> 0; (ii) one of the subgroups {g \in G : 2g=0 and {g \in G : 3g=0} is open in G; (iii) G contains an open compact subgroup of the form Z_2^κor Z_3^κfor some cardinal κ.

A Polish group containing a Haar null $F_σ$-subgroup that cannot be enlarged to a Haar null $G_δ$-set

Answering a question of Elekes and Vidnyánszky, we construct a Polish meta-abelian group $H$ and a subgroup $F\subset H$, which is a Haar null $F_σ$-set in $H$ that cannot be enlarged to a Haar null $G_δ$-set.

Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

In the paper we would like to pay attention to some analogies between Haar meager sets and Haar null sets. Among others, we will show that $0\in \inn (A-A)$ for each Borel set $A$, which is not Haar meager in an abelian Polish group. Moreover, we will give an example of a Borel non-Haar meager set $A\subset c_0$ such that $\inn (A+A)=\emptyset$. Finally, we will define $D$-measurability as a topological analog of Christensen measurability, and apply our generalization of Piccard's theorem to prove that each $D$-measurable homomorphism is continuous. Our results refer to the papers \cite{Ch}, \…

Countably compact group topologies on non-torsion abelian groups of size continuum with non-trivial convergent sequences

Under $\mathfrak{p} = \mathfrak{c}$, we answer Question 24 of \cite{dikranjan&shakhmatov3} for cardinality ${\mathfrak c}$ , by showing that if a non-torsion Abelian group of size continuum admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.

Forcing a classification of non-torsion Abelian groups of size at most $2^\mathfrak c$ with non-trivial convergent sequences

We force a classification of all the Abelian groups of cardinality at most $2^\mathfrak c$ that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and Shakhmatov for cardinality at most $2^{\mathfrak c}$, by showing that if a non-torsion Abelian group of size at most $2^\mathfrak c$ admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.