Freiheitssatz and phase transition for the density model of random groups

TL;DR

This paper investigates an analogue of Magnus' Freiheitssatz within the Gromov density model of random groups, revealing a phase transition at a specific density where the subgroup structure changes dramatically.

Contribution

It establishes a phase transition phenomenon for free subgroup generation in random groups at a critical density, extending classical results to a probabilistic setting.

Findings

Phase transition at density d_r for subgroup properties

Below d_r, certain generators form free subgroups

Above d_r, generators generate the entire group

Abstract

Magnus' Freiheitssatz states that if a group is defined by a presentation with $m$ generators and a single relator containing the last generating letter, then the first $m-1$ letters freely generate a free subgroup. We study an analogue of this theorem in the Gromov density model of random groups, showing a phase transition phenomenon at density $d_r = \min\{\frac{1}{2}, 1-\log_{2m-1}(2r-1)\}$ with $1\leq r\leq m-1$: we prove that for a random group with $m$ generators at density $d$, if $d < d_r$ then the first $r$ letters freely generate a free subgroup; whereas if $d > d_r$ then the first $r$ letters generate the whole group.