On topological properties of the formal power series substitution group

Abstract

Certain topological properties of the group $\mathcal{J}(\mathbf{k})$ of formal one-variable power series with coefficients in a commutative topological unitary ring $\mathbf{k}$ are considered. We show, in particular, that in the case of $\mathbf{k}=\mathbb{Z}$ equipped with the discrete topology, in spite of the fact that the group $\mathcal{J}(\mathbb{Z})$ has continuous monomorphisms into compact groups, it cannot be embedded into a locally compact group. In the case where $\mathbf{k}=\mathbb{Q}$ the group $\mathcal{J}(\mathbb{Q})$ has no continuous monomorphisms into a locally compact group. In the last part of the paper the compressibility property for topological groups is considered. This property is valid for $\mathcal{J}(\mathbf{k})$ for a number of rings, in particular for the group $\mathcal{J}(\mathbb{Z})$.