Some remarks on the group of formal diffeomorphisms of the line

TL;DR

This paper studies the structure of the group of formal diffeomorphisms of the line, introducing a dense subgroup with better topological properties that allows lifting finite-dimensional representations.

Contribution

It describes a canonical dense subgroup of the pro-unipotent group of formal diffeomorphisms, enabling representation liftings that are not possible in the full group.

Findings

The dense subgroup admits liftings of finite-dimensional representations.

The group consists of series with subfactorial growth of coefficients.

Provides a detailed description of the completion for the group of formal diffeomorphisms.

Abstract

Consider a strictly positively graded finitely generated infinite-dimensional real Lie algebra $\mathfrak{g}$. It has a well-defined Lie group $\overline{\mathbf{G}}$, which is an inverse limit of finite-dimensional nilpotent Lie groups (a pro-unipotent group). Generally, representations (even finite-dimensional representations) of $\mathfrak{g}$ and actions of $\mathfrak{g}$ on manifolds do not admit liftings to $\overline{\mathbf{G}}$. There is a canonically defined dense subgroup $\mathbf{G}^\circ\subset \overline{\mathbf{G}}$ with a stronger (Polish) topology, which admits lifting of representations of $\mathfrak{g}$ in finite-dimensional spaces (and, more generally, of representations of $\mathfrak{g}$ by bounded operators in Banach spaces). We describe this completion for the group $\overline{\mathbf{Diff}}$ of formal diffeomorphisms of the line, i.e., substitutions of the form $x\mapsto x+ p(x)$, where $p(x)=a_2 x^2+\dots$ are formal series, and show that the group $\mathbf{Diff}^\circ$ consists of series with subfactorial growth of coefficients.