Random triangular groups

Authors:

• Sylwia Antoniuk,
• Tomasz Łuczak,
• Jacek Świątkowski

Abstract

Let 〈S|R〉 denote a group presentation, where S is a set of n generators while R is a set of relations consisting of distinct cyclically reduced words of length three. The above presentation is called a triangular group presentation and the group it generates is called a triangular group. We study the following model Γ(n, p) of a random triangular group. The set of relations R in Γ(n, p) is chosen randomly, namely every relation is present in R independently with probability p. We study how certain properties of a random group Γ(n, p) change with respect to the probability p. In particular, we show that there exist constants c, C >0 such that if (Formula presented), then a.a.s. a random group Γ(n, p) is a free group and if (Formula presented), then a.a.s. this group has Kazhdan’s property (T). What is more interesting, we show that there exist constants c′, C′ > 0 such that if (Formula presented), then a.a.s. a random group Γ(n, p) is neither free, nor has Kazhdan’s property (T). We prove the above statements using random graphs and random hypergraphs.