Phase transition for the existence of van Kampen 2-complexes in random groups

Gromov showed that (1993) with high probability, every bounded and reduced van Kampen diagram $D$ of a random group at density $d$ satisfies the isoperimetric inequality $\mathrm{\mid }\mathrm{\partial }D\mathrm{\mid }\ge (1-2d-s)\mathrm{\mid }D\mathrm{\mid }\mathrm{\ell }$. In this article, we adapt Gruber-Mackay’s prove for random triangular groups, showing a non-reduced 2-complex version of this inequality. Moreover, for any 2-complex $Y$ of a given geometric form, we exhibit a phase transition: we give explicitly a critical density $d_c$ depending only on $Y$ such that, in a random group at density $d$, if $d<d_c$ then there is no reduced van Kampen 2-complex of the form $Y$; while if $d>d_c$ then there exists reduced van Kampen 2-complexes of the form $Y$. As an application, we show a phase transition for the $C(p)$ small-cancellation condition: for a random group at density $d$, if $d<1\mathrm{/}(p+1)$ then it satisfies $C(p)$; while if $d>1\mathrm{/}(p+1)$ then it does not satisfy $C(p)$.