On random presentations with fixed relator length

The standard (n, k, d) model of random groups is a model where the relators are chosen randomly from the set of cyclically reduced words of length k on an n-element generating set. Gromov’s density model of random groups considers the case where n is fixed, and k tends to infinity. We instead fix k, and let n tend to infinity. We prove that for all k ≥ 2 at density d > 1/2 a random group in this model is trivial or cyclic of order two, whilst for d < 1 such 2 a random group is infinite and hyperbolic. In addition we show that for d < 1/k such a random k group is free, and that this threshold is sharp. These extend known results for the triangular (k = 3) and square (k = 4) models of random groups.