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Generic pseudogroups on
$( \mathbb{C} , 0)$ and the topology of leaves
- Part of
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- Journal:
- Compositio Mathematica / Volume 149 / Issue 8 / August 2013
- Published online by Cambridge University Press:
- 01 July 2013, pp. 1401-1430
- Print publication:
- August 2013
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- Article
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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.
