Generic Pseudogroups on (C,0) and the Topology of Leaves

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 foliations possessing a unique separatrix that is given by a cusp of the form $\{y^2-x^{2k+1}=0\}$, 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.