Sur l'hyperbolicit\'e de graphes associ\'es au groupe de Cremona

TL;DR

This paper investigates the hyperbolic properties of graphs related to the Cremona group, demonstrating that a certain Wright graph is not hyperbolic while another associated graph is Gromov-hyperbolic, advancing understanding of group actions.

Contribution

It shows that the Wright graph is not Gromov-hyperbolic and introduces a new Gromov-hyperbolic graph related to the Cremona group, answering a previous open question.

Findings

Wright graph is not Gromov-hyperbolic.

Constructed a Gromov-hyperbolic graph associated with the Cremona group.

One of the new graphs is quasi-isometric to the Wright graph.

Abstract

To reinforce the analogy between the mapping class group and the Cremona group of rank $2$ over an algebraic closed field, we look for a graph analoguous to the curve graph and such that the Cremona group acts on it non-trivially. A candidate is a graph introduced by D. Wright. However, we demonstrate that it is not Gromov-hyperbolic. This answers a question of A. Minasyan and D. Osin. Then, we construct two graphs associated to a Vorono\"i tesselation of the Cremona group introduced in a previous work of the autor. We show that one is quasi-isometric to the Wright graph. We prove that the second one is Gromov-hyperbolic.