Free action of SL_2(F_p) on a sphere - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T16:31:07Z http://mathoverflow.net/feeds/question/13662 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13662/free-action-of-sl-2f-p-on-a-sphere Free action of SL_2(F_p) on a sphere Hanno Becker 2010-02-01T11:34:34Z 2010-02-02T01:41:28Z <p>Let $p>2$ be prime. Then for abstract reasons the special linear group <code>$\text{SL}_2({\mathbb F}_p)$</code> possesses a free action on some sphere (one has to check that any abelian subgroup of <code>$\text{SL}_2({\mathbb F}_p)$</code> is cyclic and that there's at most one element of order $2$).</p> <p>Does somebody know a concrete example for such a free action for general $p$? (For $p=5$, for example, <code>$\text{SL}_2({\mathbb F}_p)$</code>is the binary icosahedral group which is a subgroup of <code>${\mathbb S}^3$</code> 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).</p> http://mathoverflow.net/questions/13662/free-action-of-sl-2f-p-on-a-sphere/13767#13767 Answer by Pavel Etingof for Free action of SL_2(F_p) on a sphere Pavel Etingof 2010-02-02T01:41:28Z 2010-02-02T01:41:28Z <p>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). </p>