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I wonder whether it is possible to have a smooth action of the $ax+b$ Lie group with compactly supported fundamental vector fields on $\mathbb{R}^2$ in such a way that it is non-trivial at least at one point, i.e.linearily independent fundamtental vector fields or so.

The abelian case is possible (also in higher dimensions): there are $n$ commuting vector fields with compact support on $\mathbb{R}^n$ being linearily independent inside an open ball. So §ax+b$ would be the first step into non-commutative examples. IN higher dimensions I have examples for say the Heisenberg group.

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What's a "fundamental vector field"? Is it somehow a vector field that you can obtain pointwise as "velocity" of the flow lines on $\mathbb{R}^2$ given by the induced action of a 1-parameter subgroup? – Qfwfq Jan 27 '11 at 10:20
@unknowngoogle: yepp, that's the standard definition – Stefan Waldmann Jan 27 '11 at 10:23
up vote 9 down vote accepted

No, it's not possible, because of the generalized Reeb Stability Theorem, A generalization of the Reeb Stability Theorem, William P. Thurston, Topology, V13, pp 347--352, 1974. The theorem basically says that for any group of $C^1$-smooth diffeomorphisms of a manifold that has a fixed point where every element has first derivative trivial, the group action near that point has a generalized nilpotent structure --- the intersection of the lower central series is trivial, for some ordinal $\alpha$. (My original interest was for understanding holonomy around leaves of foliations.)

This generalized nilpotence phenomenon is fairly obvious for anything detected by the Taylor expansion at a point: if you look at the Taylor series for a vector field, commutators of vector fields with trivial $0$th and first term vanish to an even higher order. The main point is to understanding diffeomorphisms (or vector fields) that either have $C^\infity$ contact to the identity, or are not smooth enough to analyze with a Taylor series. This phenomenon is also related to the phenomenon analyzed by Margulis and others, that discrete groups of Lie groups generated by "small" elements are nilpotent.

For a Lie group, this result implies that any action near a fixed point where it has $C^1$ contact to the identity factors through a nilpotent Lie group. For a Lie group with a compactly supported action, apply this to a point on the frontier of any orbit, to conclude that the orbit factors through the action of a nilpotent quotient (in particular the group modulo the smallest term in its lower central series). For the affine group, this quotient is $\mathbb R$.

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@Bill Thurston: thanks a lot. that clarifies the situation very much. – Stefan Waldmann Jan 28 '11 at 7:49
Thanks again... – Stefan Waldmann Jan 28 '11 at 7:50

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