Quasi-isometry invariance of relative filling functions

For a finitely generated group G and collection of subgroups P , we prove that the relative
Dehn function of a pair .G; P / is invariant under quasi-isometry of pairs. Along the way, we show
quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned-off Cayley graphs. We also prove that for a cocompact simply connected combinatorial
G-2-complex X with finite edge stabilisers, the combinatorial Dehn function is well defined if and
only if the 1-skeleton of X is fine. We also show that if H is a hyperbolically embedded subgroup
of a finitely presented group G, then the relative Dehn function of the pair .G;H / is well defined. In
the appendix, it is shown that the Baumslag–Solitar group BS.k; l/ has a well-defined Dehn function
with respect to the cyclic subgroup generated by the stable letter if and only if neither k divides l
nor l divides k.