Discrete orbits and special subgroups of $\Diff$

Abstract

The local topological dynamics of subgroups of $\mathrm{Diff}({\mathbb{C}}^n,0)$, with special emphasis on $\mathrm{Diff}({\mathbb{C}}^2,0)$, is discussed with a view towards integrability questions. It is proved in particular that a subgroup of $\mathrm{Diff}({\mathbb{C}}^2,0)$ possessing locally finite orbits is necessarily solvable. Other results and examples related to higher-dimensional generalizations of Mattei-Moussu's celebrated topological characterization of integrability are also provided. These examples also settle a fundamental question raised by the previous work of Camara-Scardua.