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 2 such a random group is infinite and hyperbolic. In addition, we show that for d < 1 k such a random 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. [ABSTRACT FROM AUTHOR]