Kazhdan groups have cost 1

2018/10/25 by Tom Hutchcroft, Hutchcroft, Tom, Gábor Pete +1 · 5 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR) #math.DS #math.GR #math.PR

paper · pdf · doi:10.48550/arxiv.1810.11015

16 pages, 2 figures. V2: Minor revisions. Added additional remarks and simulations. Accepted version, to appear in Invent. Math

arxiv created 2020/04/09 · arxiv updated 2020/04/10

Abstract

We prove that every countably infinite group with Kazhdan's property (T) has cost 1, answering a well-known question of Gaboriau. It remains open if they have fixed price 1.

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• Quantitative indistinguishability and sparse and dense clusters in factor of IID percolations
• Vanishing uniqueness thresholds in Voronoi percolation on products
• Higher property T and below-rank phenomena of lattices
• Products of Infinite Countable Groups Have Fixed Price One
• Nonamenable Poisson zoo

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