Finite-valued n-isoperimetric functions across ring norms

Determine the class of ring norms under which, for any group G of type FP_{n+1}(R) equipped with an n-admissible projective RG-resolution of the trivial module R and the induced ℓ-norm, the associated n-isoperimetric function f_n(l) takes only finite values for all l ≥ 0.

Background

The paper defines the homological n-isoperimetric function for a group G over a normed ring R using an n-admissible projective resolution and the ℓ-norm induced by the ring norm. While finiteness of this function is known in certain settings (e.g., over permutation ZG-modules and for R = Z with the usual norm via [FM18]), the general situation for arbitrary norms on rings is not settled.

This question seeks to identify precisely which norms ensure the supremum-based definition of the n-isoperimetric function yields finite values, a prerequisite for quasi-isometry invariance results and for comparing isoperimetric behavior across groups and spaces.