Complete classification of the Dehn functions of Bestvina-Brady groups

TL;DR

This paper classifies the Dehn functions of all finitely presented Bestvina-Brady groups, showing they grow polynomially with degrees up to four, and provides criteria to determine this growth based on the defining graph.

Contribution

It offers a complete classification of Dehn function growth rates for Bestvina-Brady groups and links these to graph properties, also identifying obstructions to CAT(0) structures.

Findings

Dehn functions grow as linear, quadratic, cubic, or quartic polynomials.

Explicit criteria based on the defining graph determine the polynomial degree.

Certain Bestvina-Brady groups cannot admit a CAT(0) structure due to these obstructions.

Abstract

We prove that the Dehn function of every finitely presented Bestvina-Brady group grows as a linear, quadratic, cubic, or quartic polynomial. In fact, we provide explicit criteria on the defining graph to determine the degree of this polynomial. As a consequence, we identify an obstruction that prevents certain Bestvina-Brady groups from admitting a CAT(0) structure.