[Paper Review] Cocompactly cubulated 2-dimensional Artin groups

Jingyin Huang, Kasia Jankiewicz+1 more|arXiv (Cornell University)|Oct 28, 2015

Geometric and Algebraic Topology31 references3 citations

TL;DR

This paper provides a complete characterization of when 2-dimensional and three-generator Artin groups are (virtually) cocompactly cubulated, showing that such groups admit a proper, cocompact action on a CAT(0) cube complex if and only if their defining graphs satisfy a specific labeling condition: all interior edges must be labeled 2, and all leaves must be labeled with even numbers. The result resolves a key question in geometric group theory regarding cubulation of Artin groups beyond the right-angled case.

ABSTRACT

We give a necessary and sufficient condition for a 2-dimensional or a three-generator Artin group $A$ to be (virtually) cocompactly cubulated, in terms of the defining graph of $A$.

Motivation & Objective

• To determine necessary and sufficient conditions for 2-dimensional Artin groups to be (virtually) cocompactly cubulated.
• To extend this characterization to three-generator Artin groups.
• To resolve the cubulation status of prominent examples such as the 4-strand braid group.
• To clarify the role of defining graph structure in determining cubical geometry of Artin groups.
• To provide a complete classification of cocompact cubulation for this class of Artin groups.

Proposed method

• Using the theory of CAT(0) cube complexes and group actions, the authors analyze the geometry of two-generator special subgroups.
• They prove that cocompact cubulation implies convex cocompactness of these subgroups, which decompose as products of vertical and horizontal factors.
• The proof relies on a key result (Theorem 3.8) stating that a top-rank product of hyperbolic groups acting on a CAT(0) cube complex is convex cocompact.
• Algebraic and geometric arguments are used to show that if the defining graph violates the labeling condition, the intersection of subgroups cannot be either vertical or horizontal, contradicting convex cocompactness.
• The authors apply results from cubical small cancellation and the structure of fundamental groups of graph manifolds to analyze non-right-angled cases.
• They use quasi-isometric embeddings and asymptotic rank arguments in the quotient group to rule out cubulation for certain non-2-dimensional cases.

Experimental results

Research questions

• RQ1 Under what conditions on the defining graph is a 2-dimensional Artin group cocompactly cubulated?
• RQ2 When is a three-generator Artin group virtually cocompactly cubulated?
• RQ3 Why is the 4-strand braid group not virtually cocompactly cubulated?
• RQ4 What structural constraints arise in the defining graph when a 2-dimensional Artin group acts properly and cocompactly on a CAT(0) cube complex?
• RQ5 How do the labeling of edges (especially 2 and even numbers) and the presence of interior edges affect cubulation?

Key findings

• A 2-dimensional Artin group is cocompactly cubulated if and only if each connected component of its defining graph has all interior edges labeled 2 and all leaves labeled with even numbers.
• For three-generator Artin groups, cocompact cubulation holds if and only if the defining graph satisfies the same condition or has exactly two edges labeled 2.
• The 4-strand braid group is not virtually cocompactly cubulated, as its defining graph does not satisfy the required labeling condition.
• The equivalence between cocompact and virtual cocompact cubulation fails for Coxeter groups, but holds for the Artin groups studied here.
• The paper establishes that the condition in Theorem 1.1 (iii) implies cocompact cubulation even for general Artin groups, not just 2-dimensional ones.
• The authors show that if a group is cocompactly cubulated, then its two-generator special subgroups act convex cocompactly on convex subcomplexes with product structure.