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## The Groups of Basic Automorphisms of Complete Cartan Foliations

Lobachevskii Journal of Mathematics. 2018. Vol. 39. No. 2. P. 271-280.
K. I. Sheina, N. I. Zhukova.

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 e ective
then g0(M; F) = g1(M; F). We prove 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 nite-dimensional
Lie group. In particular, we obtain sucient 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.