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Fixed points of Lie group actions on surfaces

Published online by Cambridge University Press:  19 September 2008

J. F. Plante
J. F. Plante
Affiliation:
University of North Carolina, Chapel Hill, North Carolina 27514, USA
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Abstract

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Let G be a connected finite-dimensional Lie group and M a compact surface. We investigate whether, for a given G and M, every continuous action of G on M must have a fixed (stationary) point. It is shown that when G is nilpotent and M has non-zero Euler characteristic that every action of G on M must have a fixed point. On the other hand, it is shown that the non-abelian 2-dimensional Lie group (affine group of the line) acts without fixed points on every compact surface. These results make it possible to complete this investigation for Lie groups of dimension at most 3.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 1 , March 1986 , pp. 149 - 161
Copyright
Copyright © Cambridge University Press 1986

References

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