We study groups of germs of complex diffeomorphisms having a
property called irreducibility. The notion is motivated by the
similar property of the fundamental group of the complement of an
irreducible hypersurface in the complex projective space. Natural
examples of such groups of germ maps are given by holonomy groups
and monodromy groups of integrable systems (foliations) under
certain conditions. We prove some finiteness results for these
groups extending previous results in [D. Cerveau and F. Loray,
Un théorème de Frobenius singulier via l’arithmétique élémentaire,
J. Number Theory 68 1998, 2, 217–228].
Applications are given to the framework of germs of holomorphic
foliations. We prove the existence of first integrals under certain
irreducibility or more general conditions on the tangent cone of
the foliation after a punctual blow-up.