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.

This note is a companion paper to ‘Monodromie et classification topologique des germes de feuilletages holomorphes’ (in French) by Marin and Mattei. In their paper the authors develop a very complete theory about topologically conjugate singularities of foliations. However, their techniques require the topological conjugacies to be transversely holomorphic. The purpose of the present note is to show that this assumption can be made without restricting the generality of their results.