Maximal Abelian Torsion Subgroups of Diff(C,0)

TL;DR

This paper explores the structure of maximal abelian torsion subgroups within the group of germs of diffeomorphisms fixing the origin in the complex plane, showing their existence and embedding properties.

Contribution

It demonstrates the existence of maximal abelian torsion subgroups in Diff(C,0) and proves that any subgroup of Q/Z can be embedded as such a subgroup.

Findings

Existence of maximal abelian torsion subgroups in Diff(C,0)

Embedding of any subgroup of Q/Z into Diff(C,0) as a maximal abelian torsion subgroup

Clarification of the relationship between centralizers and maximal abelian subgroups in local dynamics

Abstract

In the study of the local dynamics of a germ of diffeomorphism fixing the origin in C, an important problem is to determine the centralizer of the germ in the group Diff(C,0) of germs of diffeomorphisms fixing the origin. When the germ is not of finite order, then the centralizer is abelian, and hence a maximal abelian subgroup of Diff(C,0). Conversely any maximal abelian subgroup which contains an element of infinite order is equal to the centralizer of that element. A natural question is whether every maximal abelian subgroup contains an element of infinite order, or whether there exist maximal abelian torsion subgroups; we show that such subgroups do indeed exist, and moreover that any infinite subgroup of the rationals modulo the integers Q/Z can be embedded into Diff(C,0) as such a subgroup.