Cubulating random groups in the square model

Abstract

Our main result is that for densities < 3 10 a random group in the square model has the Haagerup property and is residually finite. Moreover, we generalize the Isoperimetric Inequality, to some class of non-planar diagrams and, using this, we introduce a system of modified hypergraphs providing the structure of a space with walls on the Cayley complex of a random group. Then we show that the natural action of a random group on this space with walls is proper, which gives the proper action of a random group on a CAT(0) cube complex. 1 2 TOMASZ ODRZYGÓŹDŹ Proof. The residual finiteness results from [Ago13, Corollary 1.2] and the Haagerup Property by a folklore remark (see for example [CMV04]).To obtain Theorem 1.2 we needed to generalize the isoperimetric inequality ([Oll07, Theorem 2]) to a class of non-planar diagrams (see Theorem 2.5). This generalization has already proved to be useful in other work on random groups (see for example [MP15]). Theorem 2.5 is stated in the regime of the square model, but the reasoning can be easily repeated in the Gromov model (see [Odr14] for some version of it). The main idea of the proof of Theorem 1.2 is based on introducing modified hypergraps, which provide a structure of a space with walls on the Cayley complex of a random group, that has desired metric properties. This modified walls allowed us to construct a proper action of a random group on a space with walls, which was impossible to obtain using standard hypergraphs (introduced in [OW11] for the Gromov model and in [Odr16] for the square model).Organization. The paper is organized as follows: first we prove a generalization of the isoperimetric inequality, then using this we introduce "corrected" hypergraphs that provide the structure of a space with walls on the Cayley complex of a random group. Finally we show that a random group acts properly on this space with walls. The idea of correcting hypergraphs was inspired by [MP15].

Property (T) in random quotients of hyperbolic groups at densities above 1/3

“…There is a similar model (the k-gonal model ) to the density model where we keep l fixed and let n tend to infinity. There are results concerning hyperbolicity [Ż03, Odr16, ARD20], cubulation [Duo17,Odr18,Odr19], and Property (T) [Ż03, KK13, AŁS15, Odr19, Mon22, Ash21] in this model.…”

CAT(0) cube complexes with flat hyperplanes

“…Secondly, CAT(0) cube complexes can be reconstructed from their hyperplanes, leading to easy constructions of CAT(0) cube complexes from cubulations of pocsets and spaces with walls [Sag95,Rol98,HP98,CN05b,Nic04]. Such constructions allow us to prove that many groups naturally act on CAT(0) cube complexes, including many Artin groups [CD95, GP12, CMW19], graph braid groups [Abr00], Coxeter groups [NR03], small cancellation groups [Wis04, AO15, MS17], Thompson's groups [Far03,Far05], random groups [OW11,Odr18], many 3-manifold groups [BW12, PW14, HP15, PW18, Tid18], one-relator groups with torsion [LW13], many free-by-cyclic groups [HW15,HW16], some Burnside groups [Osa18], Cremona groups [LU20]. As a consequence, looking for an action on a CAT(0) cube complex is a useful geometric strategy in order to study a given group, but it also has applications in other areas of mathematics, most famously in low-dimensional topology [Ago13].…”

Random groups do not have Property (T) at densities below 1/4

“…The results of Ollivier-Wise, Mackay-Przytycki, and Montee were proved by building separating subspaces of the Cayley complex of a random group, and then employing the cubulation techniques of Sageev [Sag95]. These subspaces were constructed inductively by joining midpoints of edges to form 'hypergraphs', using the isoperimetric inequality for random groups (see [Oll07,Odr18]) to prove that these hypergraphs are 2-sided trees. As d tends to 1/4 these methods present increasing combinatorial difficulties.…”

“…A random group in the k-angular model at density d is defined by choosing G ∼ G(n, k, d) and letting n tend to infinity. This was preceded by the triangular model of Żuk[Ż03] and the square model of Odrzygóźdź[Odr18]. The positive k-angular model, G + (n, k, d), is obtained similarly, by choosing n kd positive words (words containing no inverse letters) of length k in F n .…”