The range of thresholds for diameter 2 in random Cayley graphs

Given a group G, the model G(G, p) denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. Given a family of groups (Gk) and a c ∈ R+ we say that c is the threshold for diameter 2 for (Gk) if for any ε > 0 with high probability Γ ∈ G(Gk, p) has diameter greater than 2 if p 6 √ (c− ε) log n n and diameter at most 2 if p > √ (c + ε) log n n . In [5] we proved that if c is a threshold for diameter 2 for a family of groups (Gk) then c ∈ [1/4, 2] and provided two families of groups with thresholds 1/4 and 2 respectively. In this paper we study the question of whether every c ∈ [1/4, 2] is the threshold for diameter 2 for some family of groups. Rather surprisingly it turns out that the answer to this question is negative. We show that every c ∈ [1/4, 4/3] is a threshold but a c ∈ (4/3, 2] is a threshold if and only if it is of the form 4n/(3n− 1) for some positive integer n.