Maximal abelian torsion subgroups of Diff(<b> C</b>,0)

Abstract

In the study of the local dynamics of a germ of diffeomorphismfixing the origin in $\mathbb C$, an important problem is to determinethe centralizer of the germ in the group Diff$(\mathbb C,0)$ of germsof diffeomorphisms fixing the origin. When the germ is not offinite order, then the centralizer is abelian, and hence a maximalabelian subgroup of Diff$(\mathbb C,0)$. Conversely any maximalabelian subgroup which contains an element of infinite order isequal to the centralizer of that element. A natural question iswhether every maximal abelian subgroup contains an element ofinfinite order, or whether there exist maximal abelian torsionsubgroups; we show that such subgroups do indeed exist, andmoreover that any infinite subgroup of the rationals modulo theintegers $\mathbb{Q/Z}$ can be embedded into Diff$(\mathbb C,0)$ as such asubgroup.