Quasi-isometry invariance of relative filling functions (with an appendix by Ashot Minasyan)

For a finitely generated group G and collection of subgroups \mathcal{P} , we prove that the relative Dehn function of a pair (G,\mathcal{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 \mathrm{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 .