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Finitude géométrique en géométrie de Hilbert + an erratum/addendum

Pierre-Louis Blayac, Mickaël Crampon, Ludovic Marquis

Abstract

The paper is divided in 2 parts. The first part is the original paper of the second and third authors arXiv:1202.5442v2. The second part is an erratum/addendum written in english and concatenated at the end of the former paper. In the erratum/addentum, we amend Theorems 1.3 and 1.11 of arXiv:1202.5442v2: Finitude géométrique en géométrie de Hilbert. We seize the opportunity to show that in round Hilbert geometry, geometrical finiteness (gf) is equivalent to cusp-uniform action and to fill some small gaps that appear in two other proofs of arXiv:1202.5442v2.

Finitude géométrique en géométrie de Hilbert + an erratum/addendum

Abstract

The paper is divided in 2 parts. The first part is the original paper of the second and third authors arXiv:1202.5442v2. The second part is an erratum/addendum written in english and concatenated at the end of the former paper. In the erratum/addentum, we amend Theorems 1.3 and 1.11 of arXiv:1202.5442v2: Finitude géométrique en géométrie de Hilbert. We seize the opportunity to show that in round Hilbert geometry, geometrical finiteness (gf) is equivalent to cusp-uniform action and to fill some small gaps that appear in two other proofs of arXiv:1202.5442v2.

Paper Structure

This paper contains 70 sections, 17 theorems, 90 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Présentation des résultats
  3. Autres travaux sur le sujet
  4. Plan de l'article
  5. Géométrie de Hilbert
  6. Distance et volume
  7. Fonctions de Busemann et horosphères
  8. Dualité
  9. Le théorème de Benzécri
  10. Classification des automorphismes
  11. Le théorème de classification
  12. Petites dimensions
  13. Quelques lemmes
  14. Démonstration du théorème de classification \ref{['classi_dym_1']}
  15. Sous-groupes nilpotents discrets de $\textrm{Aut}(\Omega)$
  16. ...and 55 more sections

Key Result

Theorem B.1.1

CM2014finitude For any $\Gamma$ acting on a round convex $\Omega$, the assertions item:GF_on_Omega, item:region_parabolique, item:gf_cusps_lobat, item:thick_part, item:non_cusp_part , item:convex_core${}_1$ and (item:GF_on_boundary$\&$item:convex_core_ghyp) are equivalent. Morever they all imply ite

Figures (23)

  • Figure 1: La distance de Hilbert
  • Figure 2: Convexe-cocompact
  • Figure 3: La distance de Hilbert et la norme de Finsler
  • Figure 4: Isométries hyperbolique et parabolique
  • Figure 5: Dégénérescence des boules
  • ...and 18 more figures

Theorems & Definitions (93)

  • proof
  • proof
  • proof
  • proof : Démonstration du théorème \ref{['classi_dym_1']}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Démonstration du lemme \ref{['dom_disc']}
  • ...and 83 more