A sharper threshold for random groups at density one-half

TL;DR

This paper investigates the properties of random groups at the critical density of 1/2, revealing a nuanced phase transition where hyperbolic groups become generic in certain regimes, extending prior understanding of the threshold behavior.

Contribution

It provides a detailed analysis of the phase transition at density 1/2, showing that hyperbolic groups are generic in some regimes, and includes an improved exposition of Kozma's unpublished argument.

Findings

Hyperbolic groups are generic below a certain growth rate at d=1/2.

Trivial groups are generic for higher growth rates at d=1/2.

The paper refines the understanding of the phase transition at the critical density.

Abstract

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 (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)