[Paper Review] Effective quasimorphisms on right-angled Artin groups

Talia Fernós, Max Forester+1 more|arXiv (Cornell University)|Feb 18, 2016

Geometric and Algebraic Topology3 citations

TL;DR

This paper constructs effective quasimorphisms on right-angled Artin groups (RAAGs) and other groups acting on CAT(0) cube complexes, using tightly nested segments of half-spaces to define quasimorphisms with uniformly bounded defect (≤12). These quasimorphisms detect all hyperbolic elements, leading to the key result that every non-trivial element in a RAAG has stable commutator length at least 1/24.

ABSTRACT

We construct new families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they "see" all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24. The group actions for which these results apply include the standard actions of right-angled Artin groups on their associated CAT(0) cube complexes. In particular, every non-trivial element of a right-angled Artin group has stable commutator length at least 1/24. These results make use of some new tools that we develop for the study of group actions on CAT(0) cube complexes: the essential characteristic set and equivariant Euclidean embeddings.

Motivation & Objective

• To develop efficient and effective quasimorphisms on groups acting on CAT(0) cube complexes with uniformly small defect.
• To establish a uniform lower bound for stable commutator length (scl) across all non-trivial elements in right-angled Artin groups.
• To introduce new tools—essential characteristic sets and equivariant Euclidean embeddings—for analyzing group actions on CAT(0) cube complexes.
• To characterize a class of group actions (RAAG-like) for which hyperbolic elements are detected by quasimorphisms.
• To extend the result to fundamental groups of special cube complexes via embedding into RAAGs.

Proposed method

• Define counting quasimorphisms φγ based on non-overlapping translates of tightly nested segments γ in the half-space structure of a CAT(0) cube complex.
• Use the median property of CAT(0) cube complexes to prove that each φγ has defect at most 6, so its homogenization has defect ≤12.
• Introduce the essential characteristic set X^ess_g to analyze the action of a hyperbolic element g and identify maximal g-nested subsegments γ.
• Apply Bavard Duality: if a homogeneous quasimorphism φ satisfies φ(g) ≥ 1 and has defect ≤12, then scl(g) ≥ 1/24.
• Use geometric arguments involving quadrant facing directions (northwest/southeast) and overlap avoidance to show that g-nested segments γ detect g via φγ.
• Prove that no translate of the reverse segment ḡγ can lie in the positive characteristic set A⁺_g, ensuring φγ(g) = 1.

Experimental results

Research questions

• RQ1 Can effective quasimorphisms with uniformly bounded defect be constructed on groups acting on CAT(0) cube complexes?
• RQ2 What conditions on group actions ensure that all hyperbolic elements are detected by such quasimorphisms?
• RQ3 Is there a uniform lower bound for stable commutator length across all non-trivial elements in right-angled Artin groups?
• RQ4 How do the new tools—essential characteristic sets and equivariant Euclidean embeddings—facilitate the construction of quasimorphisms?
• RQ5 To what extent do RAAG-like actions generalize to other groups, such as fundamental groups of special cube complexes?

Key findings

• The constructed quasimorphisms φγ have defect at most 6, so their homogenizations have defect at most 12, independent of the length of γ or the dimension of the cube complex.
• For every hyperbolic element g in a group with a RAAG-like action on a CAT(0) cube complex, there exists a quasimorphism φγ such that φ̂γ(g) ≥ 1.
• By Bavard Duality, this implies that scl(g) ≥ 1/24 for every hyperbolic element g in such groups.
• Since all non-trivial elements of a right-angled Artin group act hyperbolically on their associated CAT(0) cube complex, Corollary B establishes that scl(g) ≥ 1/24 for all non-trivial g in any RAAG.
• The result extends to fundamental groups of special cube complexes, as they embed into RAAGs and inherit the same scl lower bound.
• The proof relies on a contradiction argument involving quadrant facing directions (northwest vs. southeast) and the non-overlap of translated segments, which is forbidden by Lemma 8.3.