Critical densities for random quotients of hyperbolic groups

Ollivier, Yann

Comptes Rendus. Mathématique, Tome 336 (2003) no. 5, pp. 391-394.

Nous prouvons que pour plusieurs modèles naturels de quotient aléatoire d'un groupe, dépendant d'un paramètre de densité, pour chaque groupe hyperbolique il existe une densité critique sous laquelle un quotient aléatoire reste hyperbolique avec grande probabilité, tandis qu'au-dessus de cette densité le quotient aléatoire est très probablement trivial. Nous donnons des caractérisations explicites de ces densités critiques dans les différents modèles.

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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