Random Triangular Groups

Abstract

Let $\langle S\vert R\rangle$ 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 $\Gamma (n,p)$ of a random triangular group. The set of relations R in $\Gamma (n,p)$ is chosen randomly, namely every relation is present in R independently with probability p. We study how certain properties of a random group $\Gamma (n,p)$ change with respect to the probability p. In particular, we show that there exist constants c, C > 0 such that if $p < \frac{c} {n^{2}}$, then a.a.s. a random group $\Gamma (n,p)$ is a free group and if $p > C \frac{\log n} {n^{2}}$, 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 $\frac{C'} {n} < p < c' \frac{\log n} {n^{2}}$, then a.a.s. a random group $\Gamma (n,p)$ is neither free, nor has Kazhdan’s property (T). We prove the above statements using random graphs and random hypergraphs.

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