Random Quotients of Free Products
Eduard Einstein, Suraj Krishna M S, MurphyKate Montee, Thomas Ng, and, Markus Steenbock

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 , 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 , with the random quotient being a finite group above this density. If the density is below , the random quotient is cubulated relative to the free factors. Moreover, if the free factors are cubulated, then so is the random quotient.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · Mathematical Dynamics and Fractals
