Quasi-Isometry Invariance of discrete Higher Filling Functions

TL;DR

This paper proves that discrete higher filling functions are invariant under quasi-isometries for groups of type FP_n, confirming a conjecture and extending invariance results to weighted filling functions.

Contribution

It establishes quasi-isometry invariance of homological filling functions over discrete norms for all groups of type FP_n, confirming a key conjecture.

Findings

Homological filling functions are quasi-isometry invariants.

Invariance holds for weighted versions of filling functions.

Technique involves equipping chain complexes with geometric structures.

Abstract

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.