Freiheitssatz and phase transition for the density model of random groups

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 $1\le r\le m-1$: 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.