Reflection groups and polytopes over finite fields, III

When the standard representation of a crystallographic Coxeter group Γ is reduced modulo an odd prime p, one obtains a finite group Gp acting on some orthogonal space over Zp. If Γ has a string diagram, then Gp will often be the automorphism group of a finite abstract regular polytope. In parts I and II we established the basics of this construction and enumerated the polytopes associated to groups of rank at most 4, as well as all groups of spherical or Euclidean type. Here we extend the range of our earlier criteria for the polytopality of Gp. Building on this we investigate the class of ‘3-infinity’ groups of general rank, and then complete a survey of those locally toroidal polytopes which can be described by our construction.