Products of Infinite Countable Groups Have Fixed Price One

We prove that the product of any two infinite countable groups has fixed price one. This resolves a longstanding problem posed by Gaboriau. The proof uses the propagation method to construct a Poisson horoball process as a weak limit of a sequence of factors of iid. We then construct a low-cost graphing by showing that the resulting horoballs have a variant of the infinite touching property almost surely, if the metric and the other parameters of the construction are chosen carefully. A novelty is providing direct simple proofs that do not rely on sophisticated results like amenability and double-recurrence, which are used in related works. An essential tool for avoiding any growth conditions is the convergence in the sense of point processes of pointed closed subsets, which is a notion from stochastic geometry. Also, to manage the overlapping of the horoballs, a generalization of the induction lemma is presented for random multisets of a group.