Helena Reis

Generic pseudogroups on and the topology of leaves

We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about

Cyclic stabilizers and infinitely many hyperbolic orbits for pseudogroups on

0 \in \mathbb{C}

Equivalence and semi-completude of foliations ☆

is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular in the classical and simplest case arising from local foliations possessing a unique separatrix that is given by a cusp of the form

Separatrices for

\{y^2-x^{2k+1}=0\}

\mathbb{C}^2

, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves and that this countable set is, indeed, infinite.

actions on 3-manifolds

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 Arnolds singularities of type A^{2n+1}.