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Let $G={\rm Diff}_0^c(\mathbf R^n)$, $n\geq 1$, be the group of compactly supported diffeomorphisms isotopic to the identity through compactly supported isotopies.

Question: Is there an example to two nontrivial diffeomorphisms $f,g\in G$ such that $f$ cannot be expressed as a product of up to two conjugates (in $G$) of $g^{\pm 1} $?

Background: It follows from the work of Burago-Ivanov-Polterovich and Tsuboi that given two nontrivial diffeomorphisms as above, one can be expressed as a product of up to six conjugates of the other. I am asking for an example where one needs at least three (two is easy).

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  • $\begingroup$ compactly isotopic means that the isotopy is the identity outside a compact subset? (so $f$ and $g$ should be equal outside a compact subset?) $\endgroup$
    – YCor
    Commented Jul 11, 2017 at 13:08
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    $\begingroup$ Yes, both of them are compactly supported (=equal to the identity outside a compact subset). $\endgroup$
    – Jarek Kędra
    Commented Jul 11, 2017 at 13:10
  • $\begingroup$ In the edited post you still don't say that $f$ and $g$ are compactly supported. (Also, in 2, by conjugate, do you mean conjugate by a compactly supported diffeomorphism?) And also "nontrivial" means "not equal to the identity"? $\endgroup$
    – YCor
    Commented Jul 11, 2017 at 13:15
  • $\begingroup$ @YCor Thanks. I hope it is clearer now. $\endgroup$
    – Jarek Kędra
    Commented Jul 11, 2017 at 13:24

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