Products of Infinite Countable Groups Have Fixed Price One

2025/09/10 by Ali Khezeli, Khezeli, Ali · 2 citations
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2509.08325

openalex publication_date 2025/09/10 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Metric criteria for fixed price of countable groups
• Unimodular random graphs with Property (T) have cost one
• Gaboriau's criterion and fixed price one for locally compact groups
• Poisson-Voronoi tessellations and fixed price in higher rank
• Ideal Poisson-Voronoi tessellations on hyperbolic spaces
• On Cheeger constants of hyperbolic surfaces
• Uniform even subgraphs and graphical representations of Ising as factors of i.i.d
• Point processes, cost, and the growth of rank in locally compact groups
• Asymptotic invariants of lattices in locally compact groups
• Kazhdan groups have cost 1
• Bernoulli actions are weakly contained in any free action

Cited by

• Infinitesimal containment and sparse factors of iid
• Quantitative indistinguishability and sparse and dense clusters in factor of IID percolations

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