Generic pseudogroups on $( \mathbb{C}, 0)$ and the topology of leaves. (August 2013) - Mattei, J.-F.
Abstract We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ 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 nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $ , our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.