Convergent sequences in iterated ultrapowers as p-compact groups

Chapter 18 - Efimov's problem

Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters

Square of countably compact groups without non-trivial convergent sequences

S2 and the Fréchet property of free topological groups

ABSTRACT. Two countable spaces having no nontrivial convergent sequences are constructed. One space has every point biaccessible (by a countable discrete set), and the other has every point accessible but not biaccessible (by a countable discrete set). It is shown that in a compact T2‐space if a set A is not closed, then there exists a free sequence contained in A whose closure is not contained in A. It follows that in a compact T2‐space of countable tightness, every nonisolated point is biaccessible.

We construct, in $\mathsf{ZFC}$, a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences, answering an old problem of van Douwen. As a consequence we also prove the existence of two countably compact groups $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ such that the product $\mathbb{G}_{0} \times \mathbb{G}_{1}$ is not countably compact, thus answering a classical problem of Comfort.

<p style='text-indent:20px;'>In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula> action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, the centralizer of any minimal aperiodic <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-action on the Cantor set is a subgroup of the centralizer of a minimal <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula>-action.

Precompact groups and convergence

We show that if a countable group G G is the free product of infinite abelian groups, then for every free, probability-measure-preserving (p.m.p.) action of G G , its orbit equivalence class is weakly dense in the space of p.m.p. actions of G G . This extends Lewis Bowen’s result for free groups.

In a paper entitled cocycles and spectra Helson and Par ry prove that for every T there exists a q~ such that V ~ has Lebesgue spectrum, moreover q~ can be chosen to be real, i.e., taking values + 1 and 1. The purpose of this paper is to extend this result to certain actions of countable groups which includes ergodic non-singular actions of countable abelian groups. We blend the method of Helson and Parry with the notions of weak equivalence and weak von Neumann transformations. In section 3 and 4 we discuss these results in connection with systems of imprimitivity. The problem of extending the result quoted above to countable groups was raised by H. Helson to one of us. It is a pleasure to acknowledge his interest and encouragement in this work.

A group G is S-universal if every countable group can be embedded in a quotient of G.In this paper we give conditions under which HNN groups are S-universal.These are then applied to showing the S-universality of certain classes of Kleinian groups (discontinuous subgroups of PSL^C)).Finally some connections are mentioned between Sg-universality and the theory of ends.Introduction.A group G is So-universal if every countable group can be embedded in a quotient of G. Thus all nonabelian free groups are Suniversal [4], while it is known that all nontrivial free products except the infinite dihedral group [10], and large classes of amalgamated free products are also , [11].Sacerdote and Schupp have extended these to certain HNN and one-relator groups [9].In this paper we give a series of results which complement those of Sacerdote and Schupp but are somewhat less technical and more directly applicable to geometrically and arithmetically defined groups [2], [3].This will be applied in 3 to showing the S?-universality of a class of Kleinian groups.Finally we make some observations on the relationship between SG;-universality and the theory of ends.1. Recall that an HNN group is a group of the form G = </" ...,t",K; rel K, t^tr1 = <P,(A)>where {L,} are subgroups of K, and {<p,} are isomorphisms L, - <p,(jL,) < K. K is called the base, {L, <p(L)} associated subgroups and </" . .., i") the free part.Groups of this type were used by Higman, Neumann and Neumann to show the S-universality of free groups of rank 2 [4].The base K can be embedded isomorphically in G, and a subgroup structure theory for HNN groups, paralleling that for amalgamated free products, has been developedby Karrass and Solitar [5].Now G has a free quotient of rank n if the free part has rank n, and since clearly a group with an S-universal quotient is itself S-universal we getLemma.An HNN group with free part of rank > 1 is SQ-universal.

We make the following three observations regarding a question popularized by Katznelson: is every subset of \({\mathbb {Z}}\) which is a set of Bohr recurrence also a set of topological recurrence? (i) If G is a countable abelian group and \(E\subseteq G\) is an \(I_0\) set, then every subset of \(E-E\) which is a set of Bohr recurrence is also a set of topological recurrence. In particular every subset of \(\{2^n-2^m : n,m\in {\mathbb {N}}\}\) which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let \({\mathbb {Z}}^{\omega }\) be the direct sum of countably many copies of \({\mathbb {Z}}\) with standard basis E. If every subset of \((E-E)-(E-E)\) which is a set of Bohr recurrence is also a set of topological recurrence, then every subset of every countable abelian group which is a set of Bohr recurrence is also a set of topological recurrence. (iii) Fix a prime p and let \({\mathbb {F}}_p^\omega \) be the direct sum of countably many copies of \({\mathbb {Z}}/p{\mathbb {Z}}\) with basis \(({\mathbf {e}}_i)_{i\in {\mathbb {N}}}\). If for every p-uniform hypergraph with vertex set \({\mathbb {N}}\) and edge set \({\mathcal {F}}\) having infinite chromatic number, the Cayley graph on \(\mathbb F_p^\omega \) determined by \(\{\sum _{i\in F}{\mathbf {e}}_i:F\in {\mathcal {F}}\}\) has infinite chromatic number, then every subset of \(\mathbb F_p^\omega \) which is a set of Bohr recurrence is a set of topological recurrence.

Let $T$ be a measure-preserving and mixing action of a countable abelian group $G$ on a probability space $(X,\mathscr S,\mu )$ and $A$ a locally compact second countable abelian group. A cocycle $c\colon G\times X\to A$ for $T$ <i>disperses</i> if $\lim_

Some primitive group rings

Combinatorial Conditions Forcing Commutativity of an Infinite Group