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I am looking for a reference to the following claims:

  1. Any compact group (connected or not) acting on $S^2$ is differentiably conjugate to a linear action. This must be classical.
  2. A circle $S^1$ acting on $RP^3$ (and supposedly any spherical space form) is differentiably conjugate to a linear action. This is probably true for every compact group acting on a $3$-dimensional spherical space form?

Wolfgang Ziller

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  • $\begingroup$ Hi Wolfgang. For (1) see Theorem 2.0 in projecteuclid.org/download/pdf_1/euclid.bams/1183552167 $\endgroup$
    – alvarezpaiva
    Commented Jan 21, 2017 at 17:54
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    $\begingroup$ According to the paper "This fairly old result combines the work of several mathematicians, including H. Poincaré and L. E. J. Brouwer (see Edmonds [37])." $\endgroup$
    – alvarezpaiva
    Commented Jan 21, 2017 at 17:54
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    $\begingroup$ In French, there is the following article by Kolev (2006) in Ens. Math.: e-periodica.ch/cntmng?pid=ens-001:2006:52::23. He attributes the result (every compact subgroup of $Homeo(S^2)$ is conjugate to a subgroup of $O(3)$) to Kerékjártó, 1941 (also in French), based on much earlier results of the same author (1934) which Kolev mentioned as "know to be confusing". He also says that the most difficult case in Kerékjártó's paper is the transitive case, which turns out to be the easiest using modern Lie group theory. $\endgroup$
    – YCor
    Commented Jan 21, 2017 at 18:50
  • $\begingroup$ @YCor I don't think this addresses the question of differentiable conjugacy. $\endgroup$
    – Igor Rivin
    Commented Jan 21, 2017 at 22:10
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    $\begingroup$ @IgorRivin indeed but certainly this is senseless and is a mistake in Question 1... at least "acting" should be specified: acting by $C^k$ transformations is a necessary condition to be $C^k$-conjugate to an isometric action. Same problem in (2). And for various $k$ (integer, $\infty$, $\omega$), this are a priori unrelated properties (I don't know if they are true). $\endgroup$
    – YCor
    Commented Jan 22, 2017 at 0:47

2 Answers 2

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For (2), see the paper of Frank Raymond (1968).

Frank Raymond, MR 219086 Classification of the actions of the circle on $3$-manifolds, Trans. Amer. Math. Soc. 131 (1968), 51--78.

For (1) there is the thesis of Harald Biller at Frankfurt (1999).

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For (1), the usual trick is to average a Riemannian metric (using Haar measure on the group) to get a fixed metric. Hence the group will be a subgroup of $SO(3)$ (e.g., by uniformization, the metric on $S^2$ is conformal to the round $S^2$, so the group must act by conformal transformations, i.e. $PSL_2(\mathbb{C})$. The maximal compact subgroup of $PSL_2(\mathbb{C})$ is $SO(3)$).

For (2), the quotient space is a 2-dimensional orbifold (maybe with boundary corresponding to the fixed locus). Away from the boundary, it will be a Seifert fibration, and one may use the classification of Seifert fibrations to see that the action is conjugate to a linear action.

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