[Paper Review] Metric systolicity and two-dimensional Artin groups

Jingyin Huang, Damian Osajda|arXiv (Cornell University)|Oct 14, 2017

Geometric and Algebraic Topology38 references3 citations

TL;DR

This paper introduces metrically systolic simplicial complexes—a new geometric framework generalizing non-positive curvature—proving that all two-dimensional Artin groups act geometrically on such complexes. This leads to new results: the Conjugacy Problem is solvable, the Dehn function is quadratic, and finitely presented subgroups inherit metric systolicity and quasi-isometric rigidity features.

ABSTRACT

We introduce the notion of metrically systolic simplicial complexes. We study geometric and large-scale properties of such complexes and of groups acting on them geometrically. We show that all two-dimensional Artin groups act geometrically on metrically systolic complexes. As direct corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: we prove that the Conjugacy Problem is solvable, and that the Dehn function is quadratic. We also show several large-scale features of finitely presented subgroups of two-dimensional Artin groups, lying background for further studies concerning their quasi-isometric rigidity.

Motivation & Objective

• To develop a new geometric framework—metrically systolic complexes—that generalizes non-positive curvature for 2-dimensional Artin groups.
• To resolve long-standing open problems about algorithmic properties of two-dimensional Artin groups, such as solvability of the Conjugacy Problem.
• To establish large-scale geometric features of finitely presented subgroups of 2D Artin groups, including filling radius and quasi-isometric embedding properties.
• To provide a non-CAT(0) alternative to existing non-positive curvature models, enabling geometric analysis in the 2-dimensional setting where CAT(0) actions fail.

Proposed method

• Introduce metrically systolic complexes via a metric condition: all essential loops in vertex links have angle length at least $2\pi$.
• Use CAT(0) disc diagrams as a key tool to analyze filling properties and curvature in metrically systolic complexes.
• Construct a modified Cayley complex for 2D Artin groups by subdividing and systolizing the standard presentation complex.
• Endow the resulting complex with a piecewise Euclidean metric such that triangles have prescribed angles ($\pi/2$, $\pi/2n$) to ensure metric systolicity.
• Apply disc diagram techniques to prove that metrically systolic complexes admit quadratic Dehn functions and satisfy Morse Lemma for 2-dimensional quasi-discs.
• Leverage the action of 2D Artin groups on the constructed complex to deduce algorithmic and geometric properties of the groups and their subgroups.

Experimental results

Research questions

• RQ1 Can a non-positive curvature-like structure be defined for 2D Artin groups that is weaker than CAT(0) but still powerful enough to derive algorithmic and geometric results?
• RQ2 Are the Conjugacy Problem and Dehn function properties solvable and quadratic, respectively, for all two-dimensional Artin groups?
• RQ3 Do finitely presented subgroups of 2D Artin groups inherit metric systolicity and exhibit quasi-isometric rigidity?
• RQ4 Can metric systolicity be used to establish quasi-isometric embeddings of abelian and solvable subgroups in 2D Artin groups?

Key findings

• All two-dimensional Artin groups act geometrically on metrically systolic complexes, establishing a new geometric model for them.
• The Dehn function of every two-dimensional Artin group is quadratic, as a consequence of the metric systolic structure.
• The Conjugacy Problem is solvable for all two-dimensional Artin groups, provided the group is torsion-free and power-conjugacy implies equality.
• Finitely presented subgroups of two-dimensional Artin groups are themselves metrically systolic, inheriting the geometric and algorithmic properties of the ambient group.
• The filling radius for 2-spherical cycles in metrically systolic complexes is constant, indicating strong large-scale geometric control.
• Abelian subgroups of two-dimensional Artin groups are quasi-isometrically embedded, and nontrivial solvable subgroups are either $\mathbb{Z}$ or virtually $\mathbb{Z}^2$.