Cyclic stabilizers and infinitely many hyperbolic orbits for pseudogroups on (C,0)

TL;DR

This paper demonstrates that generic pseudogroups generated by two local diffeomorphisms on (C,0) typically have cyclic stabilizers and infinitely many hyperbolic orbits, with applications to foliation topology.

Contribution

It establishes generic properties of pseudogroups on (C,0), including cyclic stabilizers and infinite hyperbolic orbits, and applies these results to foliation topology and singularities.

Findings

Generic pseudogroups have cyclic stabilizers at every point.

Such pseudogroups possess infinitely many hyperbolic orbits.

Applications include insights into the topology of leaves of foliations.

Abstract

Consider a pseudogroup on (C,0) generated by two local diffeomorphisms having analytic conjugacy classes a priori fixed in Diff(C,0). We show that a generic pseudogroup as above is such that every point has (possibly trivial) cyclic stabilizer. It also follows that these generic groups possess infinitely many hyperbolic orbits. This result possesses several applications to the topology of leaves of foliations and we shall explicitly describe the case of nilpotent foliations associated to Arnold's singularities of type A^{2n+1}.