[Paper Review] A topological characterization of acylindrically hyperbolic groups

Bin Sun|arXiv (Cornell University)|Jul 14, 2017

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper introduces Condition $(lacklozenge)$, a generalization of the convergence action condition, to provide a topological characterization of acylindrically hyperbolic groups. By linking dynamical group actions to acylindrical hyperbolicity, the authors prove that non-elementary convergence groups are acylindrically hyperbolic, offering a new topological criterion for identifying such groups.

ABSTRACT

We introduce the notion of a group action with Condition $(\blacklozenge)$, which generalizes the convergence condition, and give a topological characterization of acylindrically hyperbolic groups. This result can be used to prove acylindrical hyperbolicity of groups coming from dynamical actions. As an application, we prove that non-elementary convergence groups are acylindrically hyperbolic.

Motivation & Objective

• To develop a topological criterion for identifying acylindrically hyperbolic groups.
• To generalize the convergence action condition via a new Condition $(lacklozenge)$.
• To establish a link between dynamical group actions and acylindrical hyperbolicity.
• To apply the characterization to prove acylindrical hyperbolicity of non-elementary convergence groups.

Proposed method

• Introduce Condition $(lacklozenge)$ as a generalization of the convergence condition for group actions.
• Define a topological condition on group actions that captures acylindrical hyperbolicity.
• Use the topological structure of the action space to characterize acylindrically hyperbolic groups.
• Apply the characterization to dynamical systems arising from convergence groups.
• Establish that non-elementary convergence groups satisfy Condition $(lacklozenge)$.
• Conclude acylindrical hyperbolicity via the topological characterization.

Experimental results

Research questions

• RQ1 Can Condition $(lacklozenge)$ serve as a topological criterion for acylindrical hyperbolicity in group actions?
• RQ2 How does Condition $(lacklozenge)$ relate to the classical convergence condition?
• RQ3 Do non-elementary convergence groups satisfy Condition $(lacklozenge)$?
• RQ4 Can the topological characterization be used to prove acylindrical hyperbolicity of groups from dynamical actions?
• RQ5 What is the relationship between dynamical actions and acylindrical hyperbolicity in the context of Condition $(lacklozenge)$?

Key findings

• Condition $(lacklozenge)$ generalizes the convergence condition and provides a topological characterization of acylindrically hyperbolic groups.
• Acylindrically hyperbolic groups are characterized by the existence of a group action satisfying Condition $(lacklozenge)$.
• The characterization enables proving acylindrical hyperbolicity for groups arising from dynamical actions.
• Non-elementary convergence groups are shown to be acylindrically hyperbolic via the new criterion.
• The result establishes a direct link between dynamical group actions and acylindrical hyperbolicity through topological means.