Discrete orbits and special subgroups of $\Diff$

TL;DR

This paper investigates the local topological dynamics of subgroups of diffeomorphisms near the origin in complex spaces, establishing conditions under which these groups are solvable and exploring higher-dimensional integrability.

Contribution

It proves that subgroups of ${ m Diff} ({ m C}^2, 0)$ with locally finite orbits are solvable and extends topological integrability characterizations to higher dimensions.

Findings

Subgroups with finite orbits are solvable.

Provides higher-dimensional generalizations of Mattei-Moussu's results.

Answers a fundamental question by Camara-Scardua.

Abstract

The local topological dynamics of subgroups of ${\rm Diff} ({\mathbb C^n}, 0)$, with special emphasis on ${\rm Diff} ({\mathbb C^2}, 0)$, is discussed with a view towards integrability questions. It is proved in particular that a subgroup of ${\rm Diff} ({\mathbb C^2}, 0)$ 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.