Homomorphism reductions on Polish groups

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 $φ: G \rightarrow G$ such that $H = φ^{-1} (L)$. We previously showed that there is a $K_σ$ subgroup, $L$, of the countable power of any locally compact Polish group, $G$, such that every $K_σ$ subgroup of $G^ω$ 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.

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