Metric criteria for fixed price of countable groups

2025/10/06 by Erin Bevilacqua, Lewis Bowen, Bevilacqua, Erin +1 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2510.05459

openalex publication_date 2025/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish general criteria for a countable group Γ to have fixed price 1 depending on a choice of left-invariant proper metric on Γ. We apply this criterion to show that if Γ₁,Γ₂ are two countable groups satisfying a certain growth condition then Γ₁× Γ₂ has fixed price 1. For example, Γ× Γ has fixed price 1 for any countable group Γ.

Cited by

• Infinitesimal containment and sparse factors of iid
• Products of Infinite Countable Groups Have Fixed Price One

Related

• A Product-Neighbourhood Criterion for Fixed Price One
• Gaboriau's criterion and fixed price one for locally compact groups
• Metric topological groups: their metric approximation and metric ultraproducts
• Products of Infinite Countable Groups Have Fixed Price One
• Kazhdan groups have cost 1