Abstract
We prove that an effective, analytic action of a connected Lie group G on an analytic manifold M becomes free on a comeager subset of an open subset of M when prolonged to a frame bundle of sufficiently high order. We further prove that the action of G becomes free on a comeager subset of an open subset of a submanifold jet bundle over M of sufficiently high order, thereby establishing a general result that underlies Lie's theory of symmetry groups of differential equations and the equivariant method of moving frames.
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References
S. Adams, Moving frames and prolongations of real algebraic actions, preprint, University of Minnesota, arXiv:1305.5742 (2013).
S. Adams, Freeness in higher order frame bundles, preprint, University of Minnesota, arXiv:1509.01609 (2015).
S. Adams, Real analytic counterexample to the freeness conjecture, preprint, University of Minnesota, arXiv:1509.01607 (2015).
S. Adams, Discrete free Abelian central stabilizers in a higher order frame bundle, preprint, University of Minnesota, arXiv:1602.08460 (2016).
S. Adams, Generic freeness in frame bundle prolongations of C ∞ actions, preprint, University of Minnesota, arXiv:1605.06527 (2016).
L. V. Ahlfors, Complex Analysis, 2nd Edition, McGraw-Hill Book Co., New York, 1966.
É. Cartan, La Méthode du Repère Mobile, la Théorie des Groupes Continus, et les Espaces Généralisés, Exposés de Géométrie No. 5, Hermann, Paris, 1935.
M. Fels, P. J. Olver, Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127–208.
H. W. Guggenheimer, Differential Geometry, McGraw-Hill, New York, 1963.
K. Iwasawa, On some types of topological groups, Ann. Math. 50 (1949), 507–558.
A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1991.
J. L. Kelley, General Topology, Graduate Texts in Mathematics, Vol. 27, Springer-Verlag, New York, 1975.
S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, New York, 1972.
E. L. Mansfield, A Practical Guide to the Invariant Calculus, Cambridge University Press, Cambridge, 2010.
J. Melleray, T. Tsankov, Generic representations of Abelian groups and extreme amenability, Israel J. Math. 198 (2013), 129–167.
P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995.
P. J. Olver, Moving frames and singularities of prolonged group actions, Selecta Math. 6 (2000), 41–77.
P. J. Olver, Lectures on moving frames, in: Symmetries and Integrability of Difference Equations, D. Levi, P. Olver, Z. Thomova, and P. Winternitz, eds., London Math. Soc. Lecture Note Series, Vol. 381, Cambridge University Press, Cambridge, 2011, pp. 207–246.
L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.
J. C. Oxtoby, Measure and Category, Springer-Verlag, New York, 1980.
J. Pohjanpelto, personal communication, 1998.
K. Prikry, personal communication, 2015.
S. Shelah, Can you take Solovay’s inaccessible away?, Israel J. Math. 48 (1984), 1–47.
R. M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. Math. 92 (1970), 1–56.
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ADAMS, S., OLVER, P.J. PROLONGED ANALYTIC CONNECTED GROUP ACTIONS ARE GENERICALLY FREE. Transformation Groups 23, 893–913 (2018). https://doi.org/10.1007/s00031-017-9463-4
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DOI: https://doi.org/10.1007/s00031-017-9463-4