Cubulating random quotients of hyperbolic cubulated groups

TL;DR

This paper demonstrates that low-density random quotients of hyperbolic cubulated groups preserve cubulation and hyperbolicity, using cubical small-cancellation theory and growth rate comparisons.

Contribution

It introduces new results showing that such quotients remain cubulated and hyperbolic, expanding understanding of random quotients in geometric group theory.

Findings

Low-density random quotients are cubulated and hyperbolic

Hyperplane stabilizers grow exponentially slower than the group

The proof combines cubical small-cancellation theory and growth rate analysis

Abstract

We show that low-density random quotients of cubulated hyperbolic groups are again cubulated (and hyperbolic). Ingredients of the proof include cubical small-cancellation theory, the exponential growth of conjugacy classes, and the statement that hyperplane stabilizers grow exponentially more slowly than the ambient cubical group.