Pushing fillings in right-angled Artin groups

TL;DR

This paper introduces pushing maps on cube complexes modeling RAAGs to analyze filling problems, providing new bounds on divergence and Dehn functions, and introduces orthoplex groups demonstrating sharpness of these bounds.

Contribution

It develops pushing maps for RAAGs, establishes new bounds on divergence and Dehn functions, and constructs orthoplex groups to show these bounds are optimal.

Findings

k-dimensional divergence bounded by r^{2k+2}

Dehn function of certain groups bounded by V^{(2k+2)/k}

Orthoplex groups demonstrate the bounds are sharp

Abstract

We construct "pushing maps" on the cube complexes that model right-angled Artin groups (RAAGs) in order to study filling problems in certain subsets of these cube complexes. We use radial pushing to obtain upper bounds on higher divergence functions, finding that the k-dimensional divergence of a RAAG is bounded by r^{2k+2}. These divergence functions, previously defined for Hadamard manifolds to measure isoperimetric properties "at infinity," are defined here as a family of quasi-isometry invariants of groups; thus, these results give new information about the QI classification of RAAGs. By pushing along the height gradient, we also show that the k-th order Dehn function of a Bestvina-Brady group is bounded by V^{(2k+2)/k}. We construct a class of RAAGs called "orthoplex groups" which show that each of these upper bounds is sharp.