Orthogonal groups in characteristic 2 acting on polytopes of high rank

Abstract

We show that for all integers $m\ge 2$, and all integers $k\ge 2$, the orthogonal groups $\Orth^{\pm}(2m,\Fk)$ act on abstract regular polytopes of rank $2m$, and the symplectic groups $\Sp(2m,\Fk)$ act on abstract regular polytopes of rank $2m+1$.