Abelian pro-countable groups and orbit equivalence relations

Maciej Malicki¹

¹ Department of Mathematics and Mathematical Economics Warsaw School of Economics al. Niepodległości 162 02-554 Warszawa, Poland

Fundamenta Mathematicae, Tome 233 (2016) no. 1, pp. 83-99

Résumé

We study a class of abelian groups that can be defined as Polish pro-countable groups, as non-archimedean groups with a compatible two-sided invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable discrete groups, endowed with the product topology. We show that for every non-locally compact, abelian quasi-countable group $G$ there exists a closed $L \leq G$ and a closed, non-locally compact $K \leq G/L$ which is a direct product of discrete countable groups. As an application we prove that for every abelian Polish group $G$ of the form $H/L$, where $H,L \leq {\rm Iso }(X)$ and $X$ is a locally compact separable metric space (in particular, for every abelian, quasi-countable group $G$), the following holds: $G$ is locally compact iff every continuous action of $G$ on a Polish space $Y$ induces an orbit equivalence relation that is reducible to an equivalence relation with countable classes.

Détail

DOI : 10.4064/fm987-1-2016

Keywords: study class abelian groups defined polish pro countable groups non archimedean groups compatible two sided invariant metric quasi countable groups closed subdirect products countable discrete groups endowed product topology every non locally compact abelian quasi countable group there exists closed leq closed non locally compact leq which direct product discrete countable groups application prove every abelian polish group nbsp form where leq iso locally compact separable metric space particular every abelian quasi countable group following holds locally compact every continuous action polish space nbsp induces orbit equivalence relation reducible equivalence relation countable classes