Freiheitssatz and phase transition for the density model of random groups

Magnus’ Freiheitssatz [20] states that if a group is defined by presentation with m generators and single cyclically reduced relator, this relator contains the last generating letter, then first $$m-1$$ letters freely generate free subgroup. We study an analogue of theorem in Gromov density model random groups [14], showing phase transition phenomenon at $$d_r = \min \{\frac{1}{2}, 1-\log _{2m-1}(2r-1)\}$$ $$1\le r\le m-1$$ : we prove for d, $$d < d_r$$ r subgroup; whereas > whole group. This result partially answers general problem proposed 2003 [15]: existence/nonexistence non-free subgroups