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The Groups of Basic Automorphisms of Complete Cartan Foliations

Authors
  • Sheina, K. I.1
  • Zhukova, N. I.1
  • 1 National Research University Higher School of Economics, Department of Informatics, Mathematics and Computer Sciences, ul. Myasnitskaya 20, Moscow, 101000, Russia , Moscow (Russia)
Type
Published Article
Journal
Lobachevskii Journal of Mathematics
Publisher
Pleiades Publishing
Publication Date
Mar 21, 2018
Volume
39
Issue
2
Pages
271–280
Identifiers
DOI: 10.1134/S1995080218020245
Source
Springer Nature
Keywords
License
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Abstract

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 effective then g0(M,F) = g1(M,F). Weprove 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 a finite-dimensional Lie group. In particular, we obtain sufficient 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.

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