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I would like to know examples (with references, if possible) of the following:

(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear action on $S^n$ (so $G$ is not isomorphic to any subgroup of $O(n+1,\mathbb{R})$);

(2) a finite group $G$ acting effectively and smoothly on $\mathbb{R}^n$ (any $n$) but admitting no effective action on $\mathbb{R}^n$ by affine transformations;

(3) a finite group $G$ acting effectively and smoothly on the closed n-dimensional disk $D^n$ (any $n$), but admitting no effective linear action on $D^n$ (so $G$ is not isomorphic to any subgroup of $O(n,\mathbb{R})$).

Some comments on these questions:

A paper of Petrie ("Free metacyclic group actions on homotopy spheres") gives an example of finite group acting freely on a sphere $S^n$ but admitting no free linear action on $S^n$.

For $p$-groups, there are no examples of (1) and (3), by a theorem of Dotzel and Hamrick.

An example of (3) is automatically an example of (2), since a finite group of affine transformations of ${\mathbb R}^n$ fixes a point and hence can be identified with a finite subgroup of $O(n)$.

There are in the litterature several results implying that there are no low dimensional examples of (1), (2) and (3).

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  • $\begingroup$ Could you be more specific about "no low-dimensional examples"? What is the largest dimension for which you know there are no examples? $\endgroup$
    – YCor
    Jul 16, 2015 at 20:24
  • $\begingroup$ The results in N.P. Buchdahl, S. Kwasik, R. Schultz, One fixed point actions on low-dimensional spheres. Invent. Math. 102 (1990), no. 3, 633–662, imply that there are no examples of (2) and (3) in dimensions up to 3. The results in J. Dinkelbach, B. Leeb, Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds, Geom. Topol. 13 (2009), 1129–1173 imply that there is no example of (1) in dimensions up to 3. $\endgroup$ Jul 17, 2015 at 9:36
  • $\begingroup$ A recent paper of Chen, Kwasik and Schultz (arxiv.org/abs/1412.5901) proves that there are no examples of (1) in dimension 4. $\endgroup$ Jul 17, 2015 at 16:23

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