The Groups of Basic Automorphisms of Complete Cartan Foliations

Abstract

For a complete Cartan foliation (M,F) we introduce two algebraic invariants g₀(M,F) and g₁(M,F) which we call structure Lie algebras. If the transverse Cartan geometry of (M,F) is effective then g₀(M,F) = g₁(M,F). Weprove that if g₀(M,F) is zero then in the category of Cartan foliations the group of all basic automorphisms of the foliation (M,F) admits a unique structure of a finite-dimensional Lie group. In particular, we obtain sufficient conditions for this group to be discrete. We give some exact (i.e. best possible) estimates of the dimension of this group depending on the transverse geometry and topology of leaves. We construct several examples of groups of all basic automorphisms of complete Cartan foliations.