Motivated by questions of deformations/moduli in foliation theory, we
investigate the structure of some groups of diffeomorphisms preserving a
foliation. We give an example of a C∞ foliation whose diffeomorphism
group is not a Lie group in any reasonable sense. On the positive side, we
prove that the automorphism group of a transversely holomorphic foliation or a
riemannian foliation is a strong ILH Lie goup in the sense of Omori
