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

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