Sharp phase transition theorems for hyperbolicity of random groups

2003/01/17 by Yann Ollivier, Ollivier, Yann · 1 citation
Mathematics · #20F67 #20P05 (Primary) 20F06 #60B99 (Secondary) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Probability (math.PR) #math.GR #math.PR #msc:20F06 #msc:20F67 #msc:20P05 #msc:60B99

paper · pdf · doi:10.48550/arxiv.math/0301187

91 pages ; 3rd version: improved redaction, corrected typos

openalex publication_date 2003/01/17 · arxiv created 2004/01/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that in various natural models of a random quotient of a group, depending on a density parameter, for each hyperbolic group there is some critical density under which a random quotient is still hyperbolic with high probability, whereas above this critical value a random quotient is very probably trivial. We give explicit characterizations of these critical densities for the various models.

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