A sharper threshold for random groups at density one-half. (English) Zbl 1394.20044

Duchin, Moon; Jankiewicz, Kasia; Kilmer, Shelby C.; Lelièvre, Samuel; Mackay, John M.; Sánchez, Andrew P.

Groups Geom. Dyn. 10, No. 3, 985-1005 (2016).

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 (We include an exposition of Kozma’s previously unpublished argument, with slightly improved results, for completeness.)

MSC:

20P05	Probabilistic methods in group theory
20F06	Cancellation theory of groups; application of van Kampen diagrams
20F67	Hyperbolic groups and nonpositively curved groups

Keywords:

random groups; density