Centralisers of formal maps

TL;DR

This paper investigates the structure of centralisers of formal maps tangent to the identity, revealing that in characteristic zero they contain a copy of the additive group, while in finite characteristic they contain an uncountable abelian subgroup.

Contribution

It characterizes the centralisers of formal maps tangent to the identity over different characteristics, highlighting their algebraic structures and differences.

Findings

In characteristic zero, centralisers contain a copy of (K,+).

In finite characteristic, centralisers contain an uncountable abelian subgroup.

Different proof techniques are used for finite and zero characteristic cases.

Abstract

We consider formal maps in any finite dimension $d$ with coefficients in an integral domain $K$ with identity. Those invertible under formal composition form a group $\mathcal{G}$. We consider the centraliser $C_g$ of an element $g\in\mathcal{G}$ which is tangent to the identity of $\mathcal{G}$. Elements of finite order always have an uncountable centraliser. If $g$ has infinite order and $K$ is a field of characteristic zero we show that $C_g$ contains an isomorphic copy of the additive group $(K,+)$. If $g$ has infinite order and $K$ has finite characteristic we show that $C_g$ contains an uncountable abelian subgroup. The proofs are quite different in finite characteristic and in characteristic zero, but are connected by so-called sum functions.