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-infinity 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.

Automated summary

The structure of diffeomorphisms preserving a foliation is studied, and an example of a C-infinity foliation is given whose diffeomorphism group does not have a natural Lie group structure. On the positive side, it is proven that the automorphism group of a transversely holomorphic foliation or a Riemannian foliation is a strong ILH Lie group in the sense of Omori. The relationship between these considerations and deformation problems in foliation theory is also investigated.