# On topological properties of the formal power series substitution group

### I. Babenko

Université Montpellier II, France### S. Bogatyi

Moscow State University (Lomonosov), Russian Federation

## 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 $\mathcal J(\mathbb Z)$.

## Cite this article

I. Babenko, S. Bogatyi, On topological properties of the formal power series substitution group. Enseign. Math. 59 (2013), no. 3/4, pp. 271–286

DOI 10.4171/LEM/59-3-3