# Quasi-isometry invariance of relative filling functions

## Author:

Hughes, S
Martínez-Pedroza, E
Saldaña, L

7 July 2021

## Last Updated:

2021-11-12T09:54:19.04+00:00

## abstract:

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 show 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 $k$.

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