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

Maximal Abelian Torsion Subgroups of Diff(C,0)