Projective reflection groups of finite covolume

TL;DR

This paper characterizes finite-volume Coxeter polytopes in Vinberg domains as quasiperfect of negative type and establishes conditions under which the Vinberg domain is the unique invariant properly convex domain for reflection groups.

Contribution

It provides a complete classification of finite-volume Coxeter polytopes in Vinberg domains and clarifies the uniqueness of the Vinberg domain for reflection groups of finite covolume.

Findings

Finite-volume Coxeter polytopes are exactly quasiperfect of negative type.

Vinberg domain is the only invariant properly convex domain under certain conditions.

Dimension at least 2 is required for the uniqueness result.

Abstract

We show that the Coxeter polytopes that have finite volume in their Vinberg domains are exactly the quasiperfect Coxeter polytopes of negative type, i.e. the Coxeter polytopes that are contained in their properly convex Vinberg domain, at the exception of some vertices that are C^1 points of the boundary. As a corollary, we show that for reflection groups \`a la Vinberg, the Vinberg domain is the only invariant properly convex domain if and only if the action is of finite covolume on the Vinberg domain and the dimension is at least 2.