Abelian pro-countable groups and orbit equivalence relations (Q2787133)

A Polish group \(G\) is \textit{quasicountable} if \(G\) is a subgroup of a direct product \(\prod_n G_n\) endowed with the product topology, where each \(G_n\) is a discrete countable group. It is proved that this is equivalent for \(G\) to be \textit{procountable}, that is, \(G\) is the inverse limit of an inverse system of discrete countable groups.NEWLINENEWLINEAs the main result it is proved that if a quasicountable Polish abelian group \(G\) is not locally compact, then there exists a closed subgroup \(L\) of \(G\) and there exists also a closed subgroup \(K\) of \(G/L\) such that \(K=\prod_n K_n\) where each \(K_n\) is a discrete infinite group.NEWLINENEWLINEFor \(X,Y\) Polish spaces and \(E,F\) equivalence relations on \(X,Y\) respectively, one says that \textit{\(E\) is (Borel) reducible to \(F\)} if there exists a Borel map \(f:X\to Y\) such that \(x\, E\, y\Leftrightarrow f(x)\, F\, f(y)\) for every \(x,y\in X\). Moreover, if \(\alpha:G\times X\to X\) is a continuous action of a Polish group \(G\) on a Polish space \(X\), let \(E_\alpha\) be the equivalence relation defined by \(x\, E_\alpha\, y\Leftrightarrow \exists g\in G, \alpha(g,x)=y\); the equivalence relation \(E_\alpha\) is called \textit{orbit equivalence relation} induced by \(\alpha\).NEWLINENEWLINEIn 1992 Kechris proved that if a Polish group \(G\) is locally compact, then every continuous action of \(G\) on a Polish space \(X\) induces an orbit equivalence relation that is reducible to an equivalence relation with countable classes, and Hjorth posed the problem on the validity of the converse implication. As an application of the main theorem it is shown that the converse implication holds true for quasicountable Polish abelian groups; indeed, if \(G\) is a quasicountable Polish abelian group, then \(G\) is locally compact if and only if every continuous action of \(G\) on a Polish space \(X\) induces an orbit equivalence relation that is reducible to an equivalence relation with countable classes. The latter result is extended to Polish abelian groups of the form \(H/L\), where \(H\) and \(L\) are subgroups of the group \(\mathrm{Iso}(Y)\) of all isometries of a Polish metric space \(Y\) endowed with the pointwise convergence topology.