Basic automorphisms of Cartan foliations covered by fibrations

Background. This work is devoted to the study of basic automorphisms $A_B(M,F)$ groups of Cartan foliations $(M,F)$ covered by fibrations, and to finding sufficient conditions for the existence of a finite-dimensional Lie group structure in $A_B(M,F)$. The class of Cartan foliations covered by fibrations is quite wide; it contains, in particular, Cartan $(X,G)$-foliations with Ehresmann connections, Cartan foliations with a vanishing transversal curvature, and Cartan foliations with integrable Ehresmann connection. Methods. We used the methods of foliated bundles and covering maps in this research. Results. We get sufficient conditions for the basic automorphism group of Cartan foliation covered by fibrations to admit a finite-dimensional Lie group structure in the category of Cartan foliations. Estimates of this group dimension are obtained. Moreover, for Cartan foliations with integrable Ehresmann connection, a method for computing groups of basic automorphisms is specified. Conclusions. The structure of basic automorphisms groups of Cartan foliations covered by fibration is determined by the structure of the global holonomy group of such foliations.