[Paper Review] Negative curvature of automorphism groups of graph products with applications to right-angled Artin groups

Anthony Genevois|arXiv (Cornell University)|Jul 2, 2018

Geometric and Algebraic Topology4 citations

TL;DR

This paper introduces the product graph $P(\Gamma, \mathcal{G})$ of a graph of groups $\Gamma \mathcal{G}$, proving it is Gromov-hyperbolic and unbounded when $\Gamma \mathcal{G}$ is not a direct sum. It establishes that the action of $\Gamma \mathcal{G}$ on $P(\Gamma, \mathcal{G})$ is acylindrical and extends to $\mathrm{Aut}(\Gamma \mathcal{G})$, leading to the proof that $\mathrm{Aut}(A_\Gamma)$ is acylindrically hyperbolic for finite, connected, square-free, non-join simplicial graphs $\Gamma$ with at least two vertices.

ABSTRACT

Given a (freely irreducible) product graph of groups $\Gamma \mathcal{G}$, we introduce and study its \emph{product graph} $P(\Gamma, \mathcal{G})$, defined as the graph whose vertices are the maximal product subgroups of $\Gamma \mathcal{G}$ and whose edges link two subgroups when they intersect non-trivially. One shows that: $(i)$ $P(\Gamma, \mathcal{G})$ is a Gromov-hyperbolic graph which is unbounded whenever $\Gamma \mathcal{G}$ is not a direct sum, $(ii)$ the action $\Gamma \mathcal{G} \curvearrowright P(\Gamma, \mathcal{G})$ satisfies a condition of acylindricity and induces a natural Nielsen-Thurston classification of the elements of $\Gamma \mathcal{G}$, $(iii)$ and this action naturally extends to an action of the automorphism group $\mathrm{Aut}(\Gamma \mathcal{G})$. By looking for WPD isometries in the automorphism group, we prove that, if $\Gamma$ is a finite, connected and square-free simplicial graph which does not decompose as a join and contains at least two vertices, then the automorphism group $\mathrm{Aut}(A_\Gamma)$ of the right-angled Artin group $A_\Gamma$ turns out to be acylindrically hyperbolic. Applications to the geometry of cyclic extensions of right-angled Artin groups are also included.

Motivation & Objective

• To define and study the product graph $P(\Gamma, \mathcal{G})$ as a tool for analyzing automorphism groups of graph products.
• To establish hyperbolicity and acylindricity of the action of $\Gamma \mathcal{G}$ on $P(\Gamma, \mathcal{G})$.
• To extend the action to $\mathrm{Aut}(\Gamma \mathcal{G})$ and use it to classify elements via Nielsen-Thurston theory.
• To prove acylindrical hyperbolicity of $\mathrm{Aut}(A_\Gamma)$ for right-angled Artin groups under specified topological constraints on $\Gamma$.

Proposed method

• Define $P(\Gamma, \mathcal{G})$ as the graph with vertices corresponding to maximal product subgroups of $\Gamma \mathcal{G}$, and edges connecting subgroups with nontrivial intersection.
• Prove that $P(\Gamma, \mathcal{G})$ is Gromov-hyperbolic and unbounded when $\Gamma \mathcal{G}$ is not a direct sum.
• Show that the action $\Gamma \mathcal{G} \curvearrowright P(\Gamma, \mathcal{G})$ satisfies acylindricity by analyzing stabilizers and uniformity of quasi-axes.
• Extend the action to $\mathrm{Aut}(\Gamma \mathcal{G})$ by lifting automorphisms to permutations of maximal product subgroups.
• Use the existence of WPD isometries in $\mathrm{Aut}(\Gamma \mathcal{G})$ acting on $P(\Gamma, \mathcal{G})$ to deduce acylindrical hyperbolicity.
• Apply the theory to right-angled Artin groups $A_\Gamma$ by verifying conditions on $\Gamma$ to ensure the required hyperbolicity and acylindricity.

Experimental results

Research questions

• RQ1 Under what conditions is the automorphism group of a graph product acylindrically hyperbolic?
• RQ2 How does the product graph $P(\Gamma, \mathcal{G})$ encode geometric and dynamical properties of $\Gamma \mathcal{G}$?
• RQ3 Can the action of $\Gamma \mathcal{G}$ on $P(\Gamma, \mathcal{G})$ be extended to an action of $\mathrm{Aut}(\Gamma \mathcal{G})$?
• RQ4 What role do WPD isometries play in establishing acylindrical hyperbolicity of $\mathrm{Aut}(A_\Gamma)$?
• RQ5 How do the topological properties of $\Gamma$—such as being finite, connected, square-free, and non-join—affect the structure of $\mathrm{Aut}(A_\Gamma)$?

Key findings

• The product graph $P(\Gamma, \mathcal{G})$ is Gromov-hyperbolic and unbounded whenever $\Gamma \mathcal{G}$ is not a direct sum.
• The action $\Gamma \mathcal{G} \curvearrowright P(\Gamma, \mathcal{G})$ is acylindrical, satisfying a uniformity condition on stabilizers of pairs of points.
• The action naturally extends to an action of $\mathrm{Aut}(\Gamma \mathcal{G})$ on $P(\Gamma, \mathcal{G})$, preserving the graph structure.
• For finite, connected, square-free simplicial graphs $\Gamma$ that are not joins and have at least two vertices, $\mathrm{Aut}(A_\Gamma)$ is acylindrically hyperbolic.
• The existence of WPD isometries in $\mathrm{Aut}(A_\Gamma)$ acting on $P(\Gamma, \mathcal{G})$ confirms acylindrical hyperbolicity of the automorphism group.
• Applications to cyclic extensions of right-angled Artin groups are derived from the geometric structure of $P(\Gamma, \mathcal{G})$ and the dynamics of the automorphism group.