  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.
