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Projective reflection groups of finite covolume

Balthazar Fléchelles, Seunghoon Hwang

TL;DR

This work characterizes Vinberg reflection groups that act with finite covolume on properly convex domains by showing that finite-volume in the Vinberg domain Ω_P occurs exactly when the Coxeter polytope P is quasiperfect and of negative type. The authors develop a comprehensive framework combining convex projective geometry, Vinberg theory, and the combinatorics of Coxeter polytopes to connect volume finiteness with the absence of negative-type faces and to establish uniqueness of the invariant convex domain. They prove that quasiperfect negative-type polytopes yield hyperbolic-type ends and that Ω_P is the maximal (and in many cases minimal) invariant domain for Γ_P, extending Marquis and related results without regularity hypotheses. The paper thus provides a precise, decomposition-friendly criterion for divisibility in convex projective geometry and clarifies when the Vinberg domain coincides with the convex hull of proximal limit points. These results sharpen our understanding of finite-volume convex projective orbifolds and the structure of their ends in relation to Coxeter data.

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 à 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.

Projective reflection groups of finite covolume

TL;DR

This work characterizes Vinberg reflection groups that act with finite covolume on properly convex domains by showing that finite-volume in the Vinberg domain Ω_P occurs exactly when the Coxeter polytope P is quasiperfect and of negative type. The authors develop a comprehensive framework combining convex projective geometry, Vinberg theory, and the combinatorics of Coxeter polytopes to connect volume finiteness with the absence of negative-type faces and to establish uniqueness of the invariant convex domain. They prove that quasiperfect negative-type polytopes yield hyperbolic-type ends and that Ω_P is the maximal (and in many cases minimal) invariant domain for Γ_P, extending Marquis and related results without regularity hypotheses. The paper thus provides a precise, decomposition-friendly criterion for divisibility in convex projective geometry and clarifies when the Vinberg domain coincides with the convex hull of proximal limit points. These results sharpen our understanding of finite-volume convex projective orbifolds and the structure of their ends in relation to Coxeter data.

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 à 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.
Paper Structure (24 sections, 26 theorems, 19 equations, 1 figure)

This paper contains 24 sections, 26 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convex projective geometry
  3. Convex subsets of $\mathop{\mathrm{\mathbb{P}}}\nolimits(V)$
  4. Faces of a convex set
  5. The Hilbert metric
  6. The proximal limit set
  7. Vinberg theory
  8. Coxeter groups
  9. Vinberg's theorem
  10. Cartan matrices
  11. Reduced and dual-reduced Vinberg representations
  12. Coxeter polytopes
  13. Types of Coxeter polytopes
  14. Indecomposable Coxeter polytopes
  15. Links of Coxeter polytopes
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $P$ be a Coxeter polytope of negative type. Then $P$ has finite volume in $\Omega_P$ if and only if $P$ is quasiperfect.

Figures (1)

  • Figure 1: Hilbert metric

Theorems & Definitions (64)

  • Theorem 1.1: see Theorem \ref{['thm:main']}
  • Remark 1.2
  • Theorem 1.3: see Theorem \ref{['thm:uniqueInvPropConvDomIffQP']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6: benoist2000
  • Definition 3.1
  • ...and 54 more