Linear isoperimetric inequality for hyperbolic groups over general normed rings

Determine which norms on rings R ensure that every word-hyperbolic group G has a linear R-homological 1-isoperimetric function with respect to the ℓ-norm induced by the ring norm on RG.

Background

The authors prove that for any ring R with the discrete norm, hyperbolic groups have linear isoperimetric functions, and this is known for Z, R, and Q with the absolute value norm. They note that linearity can be derived under additional assumptions such as symmetry |r| = |-r| and a uniform positive lower bound for |r| on nonzero elements.

It remains unclear which norms beyond these cases guarantee linear isoperimetric behavior for hyperbolic groups, motivating a general classification of ring norms with this property.