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 homomorphism reducible to L iff there is a continuous group homomorphism $\varphi : G \rightarrow G$ such that $H = \varphi ^{-1} (L)$. 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.