Quasi-isometry invariance of relative filling functions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866910782246092800 |
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| author | Hughes, Sam Martínez-Pedroza, Eduardo Saldaña, Luis Jorge Sánchez |
| author_facet | Hughes, Sam Martínez-Pedroza, Eduardo Saldaña, Luis Jorge Sánchez |
| contents | 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 $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_03355 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quasi-isometry invariance of relative filling functions Hughes, Sam Martínez-Pedroza, Eduardo Saldaña, Luis Jorge Sánchez Group Theory Geometric Topology Metric Geometry 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 $k$. |
| title | Quasi-isometry invariance of relative filling functions |
| topic | Group Theory Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2107.03355 |