Basic automorphism group of complete Cartan foliations covered by fibration

TL;DR

This paper establishes conditions under which the full basic automorphism group of a complete Cartan foliation forms a finite-dimensional Lie group, providing explicit formulas and examples, especially for foliations covered by fibrations.

Contribution

It introduces a sufficient condition for the automorphism group to have a Lie group structure and derives explicit formulas for foliations with discrete holonomy groups.

Findings

Automorphism group admits a unique finite-dimensional Lie group structure.

Explicit formula for automorphism group when holonomy is discrete.

Examples demonstrating computation of automorphism groups.

Abstract

We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the transverse Cartan geometry may not be effective. Some estimates of the dimension of this group depending on the transverse geometry are found. Further, we investigate Cartan foliations covered by fibrations. When the global holonomy group of that foliation is discrete, we obtain the explicit new formula for determining its basic automorphism Lie group. Examples of computing the full basic automorphism group of complete Cartan foliations are constructed.