Finitude g\'eom\'etrique en g\'eom\'etrie de Hilbert + an erratum/addendum

TL;DR

This paper revises and clarifies previous results on geometric finiteness in Hilbert geometry, providing corrected theorems and establishing equivalences with cusp-uniform actions.

Contribution

It offers an erratum to previous work, correcting key theorems and demonstrating the equivalence of geometric finiteness and cusp-uniform actions in Hilbert geometry.

Findings

Corrected Theorems 1.3 and 1.11 from previous work.

Established the equivalence between geometric finiteness and cusp-uniform actions.

Filled gaps in earlier proofs regarding Hilbert geometry.

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 g\'eom\'etrique en g\'eom\'etrie 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.