Collapse of random triangular groups: a closer look

Abstract

Abstract. The random triangular group Γ(n, t) is a group given by a presentation P = S|R , where S is a set of n generators and R is a random set of t cyclically reduced words of length three. The asymptotic behavior of Γ(n, t) is in some respects similar to that of widely studied density random group introduced by Gromov. In particular, it is known that if t ≤ n 3/2−ε for some ε > 0, then with probability 1 − o(1) Γ(n, t) is infinite and hyperbolic, while for t ≥ n 3/2+ε , with probability 1 − o(1) it is trivial. In this note we show that Γ(n, t) collapses provided only that t ≥ Cn 3/2 for some constant C > 0.By S|R we denote a presentation P of a group, where S is the set of generators and R the set of relations. A presentation P is a triangular presentation if the set R of relations consists of distinct cyclically reduced words of length three only (over the alphabet S ∪S −1 which consists of generators and their formal inverses). A triangular group is a group given by a triangular presentation.In the paper we investigate the properties of groups given by presentations P = S|R , where S is a set of n generators and R is a random subset of the set T of all N = 2n(4n 2 − 6n + 3) ∼ 8n 3 distinct cyclically reduced words of length three, that is words of the form abc, where a = b −1 , b = c −1 and c = a −1 . Here we consider two models of a random triangular group. In the random uniform model Γ(n, t), the set of relations R is chosen uniformly at random from the family of all N t subsets of T of size t. In the binomial model Γ(n, p), we select each element from T independently with probability p. We shall study the asymptotic properties of these two models as n → ∞. In particular, we say that for a given function t(n) [or p(n)] the group Γ(n, t(n)) [resp. Γ(n, p(n))] has a property a.a.s. (asymptotically almost surely) if the probability that the random group has this property tends to 1 as n → ∞. It should be mentioned that, as is well known in the theory of random structures, the asymptotic behavior of Γ(n, t) and Γ(n, p) can

A sharp threshold for collapse of the random triangular group

“…Unfortunately, the argument we use does not give any information on the asymptotic behaviour ofc(n). Nonetheless we strengthen the conjecture from [2] and predict thatc(n) tends to a limit.…”

“…We will now see how this lemma, together with the fact that Γ(n, p) collapses when p = n −3/2+o(1) (see either Olliver [8], or Antoniuk, Luczak andŚwiatkowski [2] and Theorem 2) implies Theorem 1.…”

“…Γ(n, p) collapses to the trivial group. However, the result of Antoniuk, Luczak andŚwiatkowski [2] and their conjecture we have just mentioned suggest that Γ(n, p) collapses more rapidly, i.e. that the collapsibility has a 'sharp' threshold.…”

“…Γ(n, p) collapses to the trivial group (his result is stated for a somewhat different, yet equivalent, model of random triangular group). Antoniuk, Luczak andŚwiatkowski [2] improved this result from one side and showed that there exists a constant C > 0 such that for p ≥ Cn −3/2 a.a.s. Γ(n, p) collapses to the trivial group.…”

On topological minors in random simplicial complexes

“…An upper bound of 1 2 √ n on this threshold for vanishing of the fundamental group has recently been shown by Korándi, Peled and Sudakov in [22]. In [2], Antoniuk, Luczak andŚwi ' atkowski use an idea similar to the one presented here to study the triviality of random triangular groups.…”