Phase transition for the existence of van Kampen \(2\)-complexes in random groups (Q6657458)

A random group is a random variable with values in a given set of groups, often constructed by group presentations with a fixed set of generators and a random set of relators (see [\textit{M. Gromov}, Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. Proceedings of the symposium held at the Sussex University, Brighton, July 14-19, 1991. Cambridge: Cambridge University Press (1993; Zbl 0841.20039)], see also [\textit{M. Gromov}, Geom. Funct. Anal. 13, No. 1, 73--146 (2003; Zbl 1122.20021)]).\N\NThe main result in the paper under review is a phase transition: given a geometric form \(Y\) of 2-complexes, the author finds a critical density \(d_{c}(Y)\) 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 interesting consequence the author exhibits phase transitions for small-cancellation conditions in random groups, providing explicitly the critical densities for the conditions \(C'(\lambda)\), \(C(p)\), \(B(p)\) and \(T(q)\).