Characterising Sobolev inequalities by controlled coarse homology and applications for hyperbolic spaces

We give a Sobolev inequality characterisation for the vanishing of a fundamental class in the controlled coarse homology of Nowak and Spakula for quasiconvex uniform spaces that support a local weak $(1,1)$-Poincar'e inequality. As applications, we consider visual Gromov hyperbolic spaces and Carnot groups.