Homomorphism reductions on Polish groups

TL;DR

This paper investigates the concept of homomorphism reducibility among subgroups of Polish groups, demonstrating a specific subgroup's universality property in locally compact cases but showing its failure in the group of increasing homeomorphisms.

Contribution

It extends the study of homomorphism reductions by identifying a universal subgroup in locally compact Polish groups and proving its non-existence in the group of increasing homeomorphisms.

Findings

Existence of a universal $K_\sigma$ subgroup in locally compact Polish groups.

Failure of this universality in the group of increasing homeomorphisms of the interval.

Clarification of the limitations of homomorphism reducibility in certain 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 $\varphi : G \rightarrow G$ 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.