A sharper threshold for random groups at density one-half

In the theory of random groups, we consider presentations with any fixed number mm of generators and many random relators of length ℓℓ, sending ℓ→∞ℓ→∞. If dd 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 dd. The signature theorem for this density model is a phase transition from triviality to hyperbolicity: for d<1/2d<1/2, random groups are a.a.s. infinite hyperbolic, while for d>1/2d>1/2, random groups are a.a.s. order one or two. We study random groups at the density threshold d=1/2d=1/2. Kozma had found that trivial groups are generic for a range of growth rates at d=1/2d=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 (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)