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Let $G$ be a Lie group acting properly on a smooth manifold $M$, and equip $M$ with a Riemannian manifold that is adapted to the foliation by orbits. The celebrated theorem of Palais is that there exist slices for the action.

Is there any work that find estimates for the maximum radii of slices in the given metric, in terms of other geometric data?

I'm particularly interested in the case when we have an open set $U$ in $M$ that is invariant, and when we can say $U$ has a global slice. Is is possible to say that if the diameter of $U$ is smaller than some quantity, then it has a global slice?

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  • $\begingroup$ In fact I think I would be happy with the case of a) the action of a compact Lie group, b) arising from a representation $\rho\colon G \to O(V)$. $\endgroup$ Aug 6, 2016 at 7:52

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Assume that the Riemannian metric is invariant. Then each slice through a point $x\in M$ can be transported along the orbit $G.x$, with the same diameter. The distance of $G.x$ to the nearest more singular point $y$ is an upper bound for the diameteter of a slice which is a geodesic disc. There is no global $\epsilon$ for all $x$ which is seen from the $U(1)$ action on $\mathbb R^2=\mathbb C$: Here the norm of $x$ is the upper bound. My intuition about this comes mainly from here.

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  • $\begingroup$ Yes, you've mentioned something similar to this before (mathoverflow.net/a/177341), but I was interested in knowing if one can estimate the distance to the nearest stratum in terms of other data, if no global constant is available. Certainly having an upper bound as you give is necessary, but is it sufficient? I'm really surprised this sort of question hasn't been addressed in the literature. $\endgroup$ Aug 6, 2016 at 22:29

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