Random Quotients of Free Products

TL;DR

This paper studies the properties of random quotients of free products of groups under a density model, revealing phase transitions and conditions for hyperbolicity and cubulation.

Contribution

It introduces a new density model for random quotients of free products and establishes phase transition thresholds for hyperbolicity and cubulation properties.

Findings

For density below 1/2, free factors embed and the quotient is relatively hyperbolic.

A phase transition occurs at density 1/2, above which the quotient becomes finite.

Below density 1/6, the quotient is cubulated, and cubulation is preserved when free factors are cubulated.

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.