A sharper threshold for random groups at density one-half

Abstract

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 (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)