A sharp threshold for collapse of the random triangular group

Abstract

Abstract. The random triangular group Γ(n, p) is the group given by a random group presentation with n generators in which every relator of length three is present independently with probability p. We show that in the evolution of Γ(n, p) the property of collapsing to the trivial group admits a very sharp threshold.

Vertex numbers of simplicial complexes with free abelian fundamental group

“…To prove the Ω(n 3/4 ) lower bound, we relate simplicial complexes to group presentations. Specifically, we define a 3-presentation as a group presentation S|R whose relations are of the form g a h b i c for g, h, i ∈ S; this is a generalization of triangular presentations as studied in [3,4,2]. Then we show that simplicial complexes give rise to 3-presentations.…”

“…Now let k = |S|, let c > 0 be a constant to be determined later, and apply Theorem 3.19 with V = φ(S), E = {{φ(g), φ(h), φ(i)} : r g a h b i c , r ∈ R ′ }, and λ = ck/n, to obtain S ′ ⊆ S such that: If there exists g ∈ S\S ′ with dim(S ′ ∪{g}) = dim S ′ , then we may replace S ′ with S ′ ∪{g}, preserving (1) and (2). Therefore, we may assume that for each g ∈ S \ S ′ , we have φ(g) ∈ span φ(S ′ ).…”

A sharper threshold for random groups at density one-half

“…The (♠) condition says that b → ∞. We show that w appears in one of these blocks surrounded by non-canceling letters is with probability > For each r ∈ R the word r[k + 1 : ℓ] is one of the 2m(2m − 1) ℓ−k−1 reduced words of length ℓ − k. We will find two relators r 1 , r 2 such that their tails match (i.e., r 1 [k + 1 : ℓ] = r 2 [k + 1 : ℓ]) but they differ in the previous letter (r 1 [k] r 2 [k]). We can conclude that w = r 1 r −1 2 reduces to a word of length 2k.…”

“…Hence we can apply Lemma 5 (with q = 2) to conclude that a.a.s. there exist r 1 ∈ R xz and r 2 ∈ R yz such that r 1 …”

“…After completing this project we learned of the 2014 preprint [1] which considers very similar threshold sharpness questions for a different model of random groups, called the triangular model, in which all relators have length three. They find a one-sided threshold for hyperbolicity and show that triviality admits a very sharp phase transition in a sense similar to our sense above.…”