Maximal Abelian Torsion Subgroups of Diff(C,0)

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.