Random triangular groups at density 1/3

TL;DR

This paper investigates the properties of random triangular groups in the binomial model, identifying thresholds for freeness and Kazhdan's property (T) as the probability parameter varies.

Contribution

It establishes precise probabilistic thresholds for when random triangular groups are free, have property (T), or are neither, advancing understanding of their phase transitions.

Findings

For p <= c/n^2, the group is almost surely free.

For p >= C log n/n^2, the group almost surely has Kazhdan's property (T).

In the intermediate range, the group is almost surely neither free nor has property (T).

Abstract

Let \Gamma(n,p) denote the binomial model of a random triangular group. We show that there exist constants c, C > 0 such that if p <= c/n^2, then a.a.s. \Gamma(n,p) is free and if p >= C log n/n^2 then a.a.s. \Gamma(n,p) has Kazhdan's property (T). Furthermore, we show that there exist constants C',c' > 0 such that if C'/n^2 <= p <= c' log n/n^2, then a.a.s. \Gamma(n,p) is neither free nor has Kazhdan's property (T).