Bestvina--Brady groups were first introduced by Bestvina and Brady for their interesting finiteness properties. In this talk, we discuss their Dehn functions, that are a notion of isoperimetric inequality for finitely presented groups and can be thought of as a "quantitative version" of finite presentability. A result of Dison shows that the Dehn function of a Bestvina--Brady group is always bounded above by a quartic polynomial.

Our main result is to compute the Dehn function for all finitely presented Bestvina--Brady groups. In particular, we show that the Dehn function of a Bestvina--Brady group grows as a polynomial of integer degree, and we present the combinatorial criteria on the graph that determine whether the Dehn functions of the associated Bestvina--Brady group is linear, quadratic, cubic, or quartic.

This is joint work with Chang and García-Mejía.