The groups of basic automorphisms of complete Cartan foliations
A great deal of (regular) foliation theory may be modelled in terms of Cartan geometries, formulating the notion of a Cartan foliation. The objective of the paper under review is to provide sufficient conditions for the group of basic automorphisms of a complete Cartan foliation to admit the structure of a finite-dimensional Lie group. This is achieved using Molino's theorem regarding the closure of leaves of a complete Cartan foliation \((M,F)\). The authors consider two Lie algebras \(\mathfrak{g}_0(M,F)\) and \(\mathfrak{g}_1(M,F)\) arising from this theorem, and show that if \(\mathfrak{g}_0(M,F)\) vanishes then the above automorphism group for \((M,F)\) is a Lie algebra. Sharp upper bounds for the dimension of this group are given, using the transverse geometry of the foliation. Also, conditions are provided for this group to be discrete.
scientific article; zbMATH DE number 6893826