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It is known that each action of a compact group on the 2-dimensional $S^2$ sphere is equivalent (=conjugated) to the linear action of a subgroup of $O(3)$ on $S^2$.

On the other hand, there exists a topological action of a finite group on $S^3$ which is not equivalent to the linear action of a subgroup of $O(4)$.

But any smooth action of a finite group on $S^3$ again is equivalent to the linear action of a subgroup of $O(4)$, according to results of Thurston and Perelman.

What about Lipschitz actions of finite groups on $S^3$? Are they equivalent to linear actions of subgroups of $O(4)$? Or there are examples of topologically wild Lipschitz actions on $S^3$?

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    $\begingroup$ when you say "equivalent" there's ambiguity. I assume you mean "conjugate", but, it might mean topologically conjugate, smoothly conjugate, Lipschitz conjugate... $\endgroup$
    – YCor
    Commented Mar 24, 2021 at 23:34
  • $\begingroup$ @YCor I use the terminology from the paper (jfdmath.sitehost.iu.edu/seminar/transformationgroupsb.pdf), see the last line on the page 3. $\endgroup$
    – Taras Banakh
    Commented Mar 25, 2021 at 4:56
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    $\begingroup$ For completeness, this refers to topological conjugacy. (And hence "wild" in the question would be "topologically wild".) $\endgroup$
    – YCor
    Commented Mar 25, 2021 at 5:56
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    $\begingroup$ Did you try to take some of the wild actions of $Z_2$ on $S^3$ and check if the construction is actually Lipschitz? These constructions are usually inductive, using a sequence of PL maps, so it would be a matter of getting a uniform Lipschitz bound. $\endgroup$
    – Moishe Kohan
    Commented Mar 26, 2021 at 17:16

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