Homomorphism reductions on Polish groups

Abstract

In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if $G$ is a Polish group and $H,L\subseteq G$ are subgroups, we say $H$ is {\em homomorphism reducible} to $L$ iff there is a continuous group homomorphism $\phi :G\to G$ such that $H={\phi }^{-1}(L)$. We previously showed that there is a $K_{\sigma }$ subgroup, $L$, of the countable power of any locally compact Polish group, $G$, such that every $K_{\sigma }$ subgroup of $G^{\omega }$ is homomorphism reducible to $L$. In the present work, we show that this fails in the countable power of the group of increasing homeomorphisms of the unit interval.