Sharp phase transition theorems for hyperbolicity of random groups

By Yann Ollivier. 2003

81 citations recorded by the source corpus.

AbstractWe 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/91b89d2edbffab5c7f64e2ead12b2d4e96eb181a

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In Geometry and Topology

1. Rufus Willett
2. Yann Ollivier
3. Martin Finn-Sell
4. Roberto Imbuzeiro Oliveira

In Mathematical Physics

1. Rufus Willett
2. Yann Ollivier
3. Martin Finn-Sell

Citations in the corpus

• Some notes on Property A (2006, background): “Gromov has written a much longer exposition of some of the ideas [Gro01], which was expanded on in [Sil03] and [ Oll04 ] (and see these papers for further references).”
• On a small cancellation theorem of Gromov (2003, methodology): “Using the techniques in [Del96] or [Oll04], the same kind of theorem should hold starting with any torsion-free hyperbolic group instead of the free group, provided that the girth of the graph is large enough w.r.t. the hyperbolicity constant, and that the labelling is quasi-geodesic.”
• Cubulating random groups at density less than 1/6 (2011, unknown): “The interest of the model is established through the following sharp phase transition theorem, proven by Gromov [Gro93] (see also [Oll04]): One of the motivations for the results in this paper is the following ([ ˙Zuk03]; see also the discussion in [Oll05b]): Theorem 1.3 (A. ˙Zuk) .”