Generic pseudogroups on ( C, 0) and the topology of leaves. (August 2013) - Mattei, J.-F.; Rebelo, J. C.; Reis, H.
Abstract We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg900t1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[0C ]]> </alternatives> 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 <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg900jm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[{ y^2 - x^2n+ 1 = 0} ]]> </alternatives>, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.