On topological properties of the formal power series substitution group

Abstract

Certain topological properties of the group $\mathcal J(\bf k)$ of formal one-variable power series with coefficients in a commutative topological unitary ring $\bf k$ are considered. We show, in particular, that in the case of $\bf k=\mathbb Z$ 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 $\bf k=\mathbb Q$ 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 $\mathcal J(\bf k)$ for a number of rings, in particular for the group $\mathcal J(\mathbb Z)$.