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.
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Beros, K.A. Homomorphism reductions on Polish groups. Arch. Math. Logic 57, 795–807 (2018). https://doi.org/10.1007/s00153-017-0606-z
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DOI: https://doi.org/10.1007/s00153-017-0606-z