Essential hyperbolic Coxeter polytopes

TL;DR

This paper introduces the concept of essential hyperbolic Coxeter polytopes, classifies a large class of such polytopes, and provides an algorithm for their classification, with implications for understanding hyperbolic reflection groups.

Contribution

It defines essential hyperbolic Coxeter polytopes, identifies a large combinatorial class containing all known high-dimensional examples, and develops an algorithm for their classification.

Findings

The class contains finitely many polytopes.

All known compact hyperbolic Coxeter polytopes of dimension ≥6 are in this class.

An effective classification algorithm is constructed and realized in 4D.

Abstract

We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimension at least 6 which are known to be essential, and prove that this class contains finitely many polytopes only. We also construct an effective algorithm of classifying polytopes from this class, realize it in four-dimensional case, and formulate a conjecture on finiteness of the number of essential polytopes.