Reflection Groups and Polytopes over Finite Fields, III

TL;DR

This paper extends the criteria for when finite groups derived from Coxeter groups over finite fields act as automorphism groups of regular polytopes, focusing on higher ranks and locally toroidal cases.

Contribution

It advances the understanding of polytopality for groups from Coxeter groups over finite fields, especially for higher ranks and specific polytope classes.

Findings

Extended criteria for polytopality of G^p groups.

Classified 3-infinity groups of general rank.

Surveyed locally toroidal polytopes from the construction.

Abstract

When the standard representation of a crystallographic Coxeter group is reduced modulo an odd prime p, one obtains a finite group G^p acting on some orthogonal space over Z_p . If the Coxeter group has a string diagram, then G^p 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 G^p . 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.