On random presentations with fixed relator length

Abstract

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.

Property (T) in k-gonal random groups

“…In order to prove his theorem, Żuk introduced a new model of random groups, called the triangular model . This was extended to the square model in [9], and in [1], this was further generalized this to a class of models, here called the k-gonal models . Roughly speaking, as k approaches infinity the behavior of k -gonal random groups should reflect the behavior of Gromov random groups.…”

“…([1]). Fix a set S of cardinality n. The k-gonal model of random groups, denoted is the set of group presentations such that R is a set of cyclically reduced words on of length k, along with the uniform probability measure.…”