Groups, Geometry, and Dynamics

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Volume 10, Issue 3, 2016, pp. 985–1005
DOI: 10.4171/GGD/374

Published online: 2016-09-16

A sharper threshold for random groups at density one-half

Moon Duchin[1], Kasia Jankiewicz[2], Shelby C. Kilmer[3], Samuel Lelièvre[4], John M. Mackay[5] and Andrew P. Sánchez[6]

(1) Tufts University, Medford, USA
(2) McGill University, Montreal, Canada
(3) University of Utah, Salt Lake City, USA
(4) Université Paris-Sud, Orsay, France
(5) University of Bristol, UK
(6) Tufts University, Medford, USA

In the theory of random groups, we consider presentations with any fixed number $m$ of generators and many random relators of length $\ell$, sending $\ell \to \infty$. 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 (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)

Keywords: Random groups, density

Duchin Moon, Jankiewicz Kasia, Kilmer Shelby, Lelièvre Samuel, Mackay John, Sánchez Andrew: A sharper threshold for random groups at density one-half. Groups Geom. Dyn. 10 (2016), 985-1005. doi: 10.4171/GGD/374