Pavage de Vorono\"i associ\'e au groupe de Cremona

TL;DR

This paper explores the action of the Cremona group of rank 2 on an infinite-dimensional hyperbolic space, constructing a Voronoi tessellation to analyze the group's geometric properties and boundary behavior.

Contribution

It introduces a Voronoi tessellation associated with the Cremona group's action, providing a geometric framework analogous to classical hyperbolic group actions.

Findings

Constructed a fundamental domain for the Cremona group's action.

Analyzed adjacency relations between Voronoi cells.

Studied boundary points at infinity and cell sharing properties.

Abstract

The action of the Cremona group of rank 2 on an infinite dimensional hyperbolic space is the main recent tool to study the Cremona group. Following the analogy with the action of PSL(2,Z) on the Poincar\'e half-plane, we exhibit a fundamental domain for this action by considering a Vorono\"i tessellation. Then we study adjacent cells to a given cell, as well as cells that share common points in the boundary at infinity.