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Abstract

We study homeomorphism groups of metrizable compactifications of $\mathbb{N}$. All of those groups can be represented as almost zero-dimensional Polishable subgroups of the group $S_\infty$. As a corollary, we show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of $S_\infty$. We prove a sufficient condition for these groups to be one-dimensional and also study their descriptive complexity. In the last section we associate with every Polishable ideal on $\mathbb{N}$ a certain Polishable subgroup of $S_\infty$ which shares its topological dimension and descriptive complexity.