Locally Toroidal Polytopes and Modular Linear Groups

TL;DR

This paper develops methods to analyze modular reductions of crystallographic Coxeter groups for composite moduli, leading to new locally toroidal polytopes with automorphism groups derived from these groups.

Contribution

It extends previous work on prime moduli to composite moduli, providing a complete description of the resulting modular polytopes for spherical and Euclidean types.

Findings

Complete classification of modular polytopes for composite moduli

Identification of new locally toroidal polytopes

Application of a modular quotient criterion

Abstract

When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d>1, one obtains a finite group G^d which is often the automorphism group of an abstract regular polytope. Building on earlier work in the case that d is an odd prime, we here develop methods to handle composite moduli and completely describe the corresponding modular polytopes when G is of spherical or Euclidean type. Using a modular variant of the quotient criterion, we then describe the locally toroidal polytopes provided by our construction, most of which are new.