On topological properties of the formal power series substitution group

TL;DR

This paper investigates the topological characteristics of the formal power series substitution group over various rings, revealing limitations on embeddings into locally compact groups and establishing the group's compressibility.

Contribution

It provides new insights into the topological structure of the substitution group over different rings, including non-embeddability and compressibility results.

Findings

$ ext{J}( ext{Q})$ has no continuous bijections into a locally compact group.

$ ext{J}( ext{Z})$ cannot be embedded into a locally compact group despite having continuous bijections into compact groups.

The group $ ext{J}( ext{Z})$ is shown to be compressible.

Abstract

Certain topological properties of the group $\J(\k)$ of all formal one-variable power series with coefficients in a topological unitary ring $\k$ are considered. We show, in particular, that in the case when $\k=\Q$ the group $\J(\Q)$ has no continuous bijections into a locally compact group. In the case when $\k=\Z$ supplied with discrete topology, in spite of the fact that the group $\J(\Z)$ has continuous bijections into compact groups, it cannot be embedded into a locally compact group. In the final part of the paper the compression property for topological groups is considered. We establish the compressibility of $\J(\Z)$.