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

TL;DR

This paper investigates phase transitions in the existence of van Kampen 2-complexes in random groups, establishing critical densities that determine their presence or absence, and applies these results to small-cancellation conditions.

Contribution

It introduces a phase transition framework for van Kampen 2-complexes in random groups and explicitly computes critical densities based on geometric forms.

Findings

Existence of a critical density $d_c$ for van Kampen 2-complexes of a given form.

Below $d_c$, such complexes do not exist; above $d_c$, they do.

Phase transition for the $C(p)$ small-cancellation condition at $d=1/(p+1)$.

Abstract

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 $|\partial D|\geq (1-2d-s)|D|\ell$. 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 $d<1/(p+1)$ then it satisfies $C(p)$; while if $d>1/(p+1)$ then it does not satisfy $C(p)$.