Pushing fillings in right‐angled Artin groups

We define a family of quasi-isometry invariants of groups called higher divergence functions ,w hich measure isoperimetric properties "at in- finity." We give sharp upper and lower bounds on the divergenc ef unctions for right-angled Artin groups, using di!erent pushing maps on the associated cube complexes. In the process, we define a class of RAAGs we call orthoplex groups ,w hich have the property that their Bestvina-Brady subgroups have hard-to-fill spheres. Our results give sharp bounds on the higher Dehn func- tions of Bestvina-Brady groups, a complete characterization of the divergence of geodesics in RAAGs, and an upper bound for filling loops at infinity in the mapping class group. This paper focuses on a new construction for right-angled Artin groups that we call a pushing map. This map can be used to study the di!erence between e"cient filling and "obstructed" filling of cycles, by pushing chains into constrained parts of the complex associated to the group. These comparative rates of filling give information about the flexibility of the geometry. We note at the outset that the main results and techniques in this paper, though their properties are stated and proved in the homological category, work just as well with homotopical definitions. Right-angled Artin groups (or RAAGs) are given by presentations in which each relator is a commutator of two generators. These groups have been studied exten- sively, and much is known about them both algebraically and geometrically. For instance, they have automatic structures and useful normal forms, and they act ge- ometrically on CAT(0) cube complexes. Many tools are available for their study, in part because these complexes contain many flats arising from mutually commuting elements (see (Cha07)). Frequently, topological invariants of RAAGs can be read o! of the defining graph, and along these lines we will relate propertie so f the graph to the filling functions and divergence functions of the groups. The divergence function grew from the desire to study the geometry of a group "at infinity" through the use of filling functions. Recall that the most basic filling function in groups is the Dehn function, which measures the area necessary to fill a closed loop by a disk; these functions have been a key part of geometric group theory since Gromov used them to characterize hyperbolic groups (Gro87) (or arguably longer, since Dehn used related ideas to find fast solutions to the word problem).