A sharper threshold for random groups at density one-half

• Published in: arXiv.org
• Main Authors: Duchin, Moon, Jankiewicz, Kasia, Kilmer, Shelby C, Lelièvre, Samuel, Mackay, John M, Sánchez, Andrew P
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 30.12.2014
• Subjects: Density; Phase transitions
• ISSN: 2331-8422

In the density model of random groups, we consider presentations with any fixed number m of generators and many random relators of length l, sending l to infinity. If d is a "density" parameter measuring the rate of exponential growth of the number of relators compared to the length of relators, then many group-theoretic properties become generically true or generically false at different values of d. The signature theorem for this density model is a phase transition from triviality to hyperbolicity: for d < 1/2, random groups are a.a.s. infinite hyperbolic, while for d > 1/2, random groups are a.a.s. order one or two. We study random groups at the density threshold d = 1/2. Kozma had found that trivial groups are generic for a range of growth rates at d = 1/2; we show that infinite hyperbolic groups are generic in a different range. (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)