Divergence of CAT(0) Cube Complexes and Coxeter Groups

Abstract

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani-Thomas that addressed the class of 2-dimensional right-angled Coxeter groups. As another application, we provide an inductive graph theoretic criteria on a right-angled Coxeter group's defining graph which allows us to recognize arbitrary integer degree polynomial divergence for many infinite classes of right-angled Coxeter groups. We also provide similar divergence results for some classes of Coxeter groups which are not right-angled.

arXiv commentFinal prepublished version. This article now appears in Algebraic & Geometric Topology

References in corpus (8)

• Asymptotic geometry of the mapping class group and Teichmueller space
• Rank rigidity for CAT(0) cube complexes
• Weak hyperbolicity of cube complexes and quasi-arboreal groups
• The simplicial boundary of a CAT(0) cube complex
• Thickness, relative hyperbolicity, and randomness in Coxeter groups
• CAT(0) spaces with polynomial divergence of geodesics
• Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries
• Global Structural Properties of Random Graphs

Cited by in corpus (7)

• Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups, with an appendix written jointly with Christopher Cashen
• A remark on thickness of free-by-cyclic groups
• Sublinearly Morse Geodesics in CAT(0) Spaces: Lower Divergence and Hyperplane Characterization
• On the coarse geometry of certain right-angled Coxeter groups
• Divergence of finitely presented subgroups of CAT(0) groups
• Divergence, thickness and hypergraph index for general Coxeter groups
• Detecting a subclass of torsion-generated groups