Gaboriau's criterion and fixed price one for locally compact groups

2023/07/21 by Sam Mellick, Mellick, Sam · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2307.11728

openalex publication_date 2023/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G₁ be a semisimple real Lie group and G₂ another locally compact second countable unimodular group. We prove that G₁ × G₂ has fixed price one if G₁ has higher rank, or if G₁ has rank one and G₂ is a p-adic split reductive group of rank at least one. As an application we resolve a question of Gaboriau showing SL(2,ℚ) has fixed price one. Inspired by the very recent work arXiv:2307.01194v1 [math.GT], we employ the method developed by the author and Miklós Abért to show that all essentially free probability measure preserving actions of groups weakly factor onto the Cox process driven by their amenable subgroups. We then show that if an amenable subgroup can be found satisfying a double recurrence property then the Cox process driven by it has cost one.

Cited by

• Products of Infinite Countable Groups Have Fixed Price One
• Unimodular random graphs with Property (T) have cost one

Related

• Point processes, cost, and the growth of rank in locally compact groups
• A Product-Neighbourhood Criterion for Fixed Price One
• Fixed points for bounded orbits in Hilbert spaces
• Metric criteria for fixed price of countable groups
• Unimodular random graphs with Property (T) have cost one