Random quotients preserve acylindrical and hierarchical hyperbolicity

TL;DR

This paper introduces a new model for random quotients of groups using random walks, demonstrating that key hyperbolic properties are preserved in the quotients with high probability, and explores the construction of exotic hyperbolic groups.

Contribution

It establishes that acylindrical and hierarchical hyperbolicity are preserved in random quotients, and constructs new hyperbolic groups with fixed point properties.

Findings

Random quotients of acylindrically hyperbolic groups remain acylindrically hyperbolic.

Hierarchical hyperbolic structures are preserved in quotients asymptotically almost surely.

Existence of common quotients with strong fixed point properties like Kazhdan's property (T).

Abstract

We propose a new model for random quotients of groups using independent random walks. In this model, we show that random quotients of acylindrical hyperbolic groups asymptotically almost surely remain acylindrically hyperbolic. Our main tools relate the theories of spinning families and projection complexes to random walks. In the presence of a hierarchical hyperbolic structure on the group, we leverage the fine control of projections to show that this structure is preserved in the quotient asymptotically almost surely. The same techniques yield that random quotients of a non-elementary hyperbolic group (relative to any finite collection of finitely generated peripheral subgroups) are asymptotically almost surely hyperbolic (relative to commensurable peripheral subgroups). Finally, we also prove that any two groups that are both acylindrically and hierarchically hyperbolic have a common quotients which is itself acylindrically and hierarchically hyperbolic. This produces "exotic" hierarchically hyperbolic groups with strong fixed point properties, such as Kazhdan's property (T).