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Let $p>2$ be prime. Then for abstract reasons the special linear group $\text{SL}_2({\mathbb F}_p)$ possesses a free action on some sphere (one has to check that any abelian subgroup of $\text{SL}_2({\mathbb F}_p)$ is cyclic and that there's at most one element of order $2$).

Does somebody know a concrete example for such a free action for general $p$? (For $p=5$, for example, $\text{SL}_2({\mathbb F}_p)$is the binary icosahedral group which is a subgroup of ${\mathbb S}^3$ thus acting freely on it by multiplication; I'd like to know if there's one single action that can be written down for all $p$ simultaneously).

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    $\begingroup$ Does acting on a sphere mean acting by orthogonal matrices, or is it just a topological action? The former is impossible, because the smallest nontrivial representation of SL_2(F_p) has dimension (p-1)/2. $\endgroup$
    – David E Speyer
    Commented Feb 1, 2010 at 13:43
  • $\begingroup$ Yes, I just mean a free topological action, not necessarily orthogonal. But concerning your remark on orthogonal actions: How do know that there cannot exist any orthogonal action from the fact on the dimension of the smallest nontrivial representation? $\endgroup$
    – Hanno
    Commented Feb 1, 2010 at 14:25
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    $\begingroup$ Could the issue lie in your phrasing "some sphere"? I read this as meaning $\mathbb{S}^n$ for some $n$; maybe David was thinking $\mathbb{S}^2$? $\endgroup$
    – Pete L. Clark
    Commented Feb 1, 2010 at 14:34
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    $\begingroup$ Hanno, could you please explain a bit what you mean by "abstract reasons"? $\endgroup$
    – Dmitri Panov
    Commented Feb 2, 2010 at 0:07
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    $\begingroup$ I've added the gt tag. I don't know the answer to either question; all I can say is that there is no single sphere upon which all SL_2(F_p) act orthogonally. This follows from a classical result by Jordan on abelian normal subgroups of finite linear groups in characteristic zero. $\endgroup$
    – algori
    Commented Feb 2, 2010 at 1:33

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Apparently, a linear free action exists only for $p=5$ (if $p\ge 5$), see paper by C. Thomas "Almost linear actions by $SL_2(p)$ on $S^{2n-1}$". There is a weaker notion of an "almost linear" action, and it seems that constructing such actions is a fairly complicated business, using state-of-the-art differential geometry and topology; see arXiv:math/9911250. It seems that simple explicit actions for higher $p$ are not expected (also, of course, the sphere must be odd dimensional, since an orinentation preserving self-map of an even dimensional sphere has a fixed point by the Lefschetz theorem).

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