On the automorphism group of foliations with geometric transverse structures

We study the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a C ^∞ foliation whose diffeomorphism group has not a natural structure of Lie group. On the positive side, we prove that the automorphism group of a transversely holomorphic foliation or a Riemannian foliation is a strong ILH Lie group in the sense of Omori. We also investigate the relationship of the previous considerations with deformation problems in foliation theory. We show that the existence of a local moduli space for a given foliation imposes strong conditions on its automorphism group. They are not fulfilled in many cases, in particular they are not fulfilled by the foliation mentioned above.