Products of Finitely-Generated Groups with a Certain Growth Condition Have Fixed Price One

Abstract: An open problem posed by Gaboriau is whether the product of any two infinite countable groups has fixed price one. We provide an affirmative answer if the two groups are finitely generated and their growths satisfy a specific condition. The proof uses the propagation method to construct a Poisson horoball process as a weak factor of i.i.d., where each horoball is equipped with a marking that depends only on the first coordinate, in an i.i.d. manner. Then, a low-cost graphing of this process is constructed using the markings of the horoballs and adding a percolation with small intensity.