Random Quotients of Free Products

TL;DR

The paper develops a density model for random quotients of free products, extending Gromov's density framework via a Bass–Serre tree construction and a finite-relator randomization on fixed translation-length elements. It shows sharp thresholds: for density $d<\tfrac{1}{2}$, the free factors embed and the quotient is relatively hyperbolic; at $d=\tfrac{1}{2}$ a phase transition occurs to finite quotients, and for $d<\tfrac{1}{6}$ the quotient is cubulated relative to the factors (and fully cubulated if the factors are). The approach combines model spaces $X_{\mathcal{R}}$ and $X_{\mathcal{R}}(\mathcal{Z})$, local and global isoperimetric inequalities, a non-planar Greendlinger-type lemma, and a relative cubulation criterion (Éinstein–Ng) to establish relatively geometric cubulations and proper, cocompact actions on CAT(0) cube complexes. The results significantly extend cubulation phenomena to random quotients of free products, providing structural insight into their hyperbolic and cubical geometry and enabling potential applications to residual properties and separability for these quotients.

Abstract

We introduce a density model for random quotients of a free product of finitely generated groups. We prove that a random quotient in this model has the following properties with overwhelming probability: if the density is below $1/2$, the free factors embed into the random quotient and the random quotient is hyperbolic relative to the free factors. Further, there is a phase transition at $1/2$, with the random quotient being a finite group above this density. If the density is below $1/6$, the random quotient is cubulated relative to the free factors. Moreover, if the free factors are cubulated, then so is the random quotient.

Figures (9)

• Figure 1: The Bass--Serre tree $T$ and the graph of groups $\Sigma$.
• Figure 2: The actions of $G_*$ on $T(\mathcal{Z})$ and $T$, respectively, induces actions of $G$ on $X_\mathcal{R}(\mathcal{Z})$ and $X_\mathcal{R}$, respectively.
• Figure 3: A non-planar diagram with three 2-cells and six connectors, each indicated in a different color. Note that the intersection of two 2-cells need not be a connector.
• Figure 4: Two possible arrangements of adjacent faces. On the left, the edge triple $(e_1, v, e_2)$ is fully contained in both adjacent faces, and $\deg_*(e_1, v, e_2) = 2$. On the right $(e_1, v, e_2)$ is partially contained in one face and fully contained in the other, and $\deg_*(e_1, v, e_2) = 2$.
• Figure 5: Two diagrams and dual graphs are shown. The two diagrams are distinct, but their undecorated dual graphs are isomorphic.

Theorems & Definitions (142)

• Definition 1
• Theorem 2
• Example 3
• Theorem 4
• Remark 5
• Remark 6
• Remark 9
• Lemma 10
• Lemma 11
• Example 12