The groups of basic automorphisms of complete Cartan foliations (Q1646302)

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.