Random triangular groups at density

Abstract

Abstract. Let Γ(n, p) denote the binomial model of a random triangular group. We show that there exist constants c, C > 0 such that if p ≤ c/n 2 , then a.a.s. Γ(n, p) is free and if p ≥ Clog n/n 2

Fundamental groups of clique complexes of random graphs

“…Clearly I(X) = I(X (2) ), that is, the isoperimetric constant I(X) depends only on the 2skeleton X (2) .…”

“…We consider below the twodimensional skeleton X (2) Γ which can be viewed as a random 2-complex. In this section, we shall examine the validity of the Whitehead conjecture for aspherical subcomplexes of X (2) Γ . Recall that for any simplicial complex K, its first barycentric subdivision K is a clique complex.…”

“…then, with probability tending to 1 as n → ∞, the clique complex X Γ is simplicially collapsible to a graph. In particular, under the above assumption the fundamental group π 1 (X Γ , x 0 ) of a random clique complex X Γ , where Γ ∈ G(n, p), is free, for any choice of the base point x 0 ∈ X Γ ; each connected component of the 2-skeleton X (2) Γ is homotopy equivalent to a wedge of circles and 2-spheres, a.a.s.…”

“…Note that in the range (1) the dimension of X Γ is 3 and the 2-skeleton X (2) Γ contains the tetrahedron and its subdivision having five vertices, a.a.s. Hence, the 2-skeleton X (2) Γ is not collapsible to a graph.…”

“…We wish to mention a recent preprint [2] where random triangular groups are studied; this class of random groups is different from the class of fundamental groups of random clique complexes although these two classes of random groups share several common features. The main result of [2] states that there exists an interval in which the random triangular group is neither free nor possesses the property T. We expect that such intermediate regime exists in the model which we study in this paper. We shall address this issue elsewhere.…”