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.