Question

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).

Answer 1

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).