Quasi-Isometry Invariance of discrete Higher Filling Functions

We prove that homological filling functions over a ring $R$ equipped with the discrete norm are quasi-isometry invariants for all groups of type $\mathrm{FP}_n$. This confirms a conjecture of Bader-Kropholler-Vankov in the case of discrete norms. The proof uses a technique of equipping free chain complexes with a geometric structure, allowing for analogues of cellular constructions in the purely algebraic setting. As a further application we prove quasi-isometry invariance for a weighted version of integral and discrete filling functions originally introduced in the study of the rapid decay property.