The k-dimensional isoperimetric function of a space captures the difficulty of filling k-spheres with (k+1)-balls in the space. Once one understands the isoperimetric functions of a space, it is interesting to study how they change when an obstruction is introduced. In this spirit, Brady and Farb introduced the notion of filling invariants at infinity, by considering the volume required to fill spheres in Hadamard manifolds, provided both the sphere and the filling are far from a fixed basepoint. I will talk about joint work with A. Abrams, N. Brady, M. Duchin and R. Young, which develops a group theoretic version of this concept, and describe our results for right-angled Artin groups and mapping class groups.