Geodesic
Quasi-isometry invariance of relative filling functions (with an appendix by Ashot Minasyan)
Sam Hughes1 orcid ; Eduardo Martínez-Pedroza2 ; Luis Jorge Sánchez Saldaña3
1University of Oxford, UK
2Memorial University of Newfoundland, St. John’s, Canada
3Universidad Nacional Autónoma de México, Mexico
Groups, geometry, and dynamics, Tome 17 (2023) no. 4, pp. 1483-1515

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DOI

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.
DOI : 10.4171/ggd/737
Classification : 20-XX, 57-XX
Mots-clés : Quasi-isometry of pairs, relative filling function, isoperimetric inequality, subgroup rigidity

Sam Hughes  1   ; Eduardo Martínez-Pedroza  2   ; Luis Jorge Sánchez Saldaña  3

1 University of Oxford, UK
2 Memorial University of Newfoundland, St. John’s, Canada
3 Universidad Nacional Autónoma de México, Mexico 
Sam Hughes; Eduardo Martínez-Pedroza; Luis Jorge Sánchez Saldaña. Quasi-isometry invariance of relative filling functions (with an appendix by Ashot Minasyan). Groups, geometry, and dynamics, Tome 17 (2023) no. 4, pp. 1483-1515. doi: 10.4171/ggd/737
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     title = {Quasi-isometry invariance of relative filling functions (with an appendix by {Ashot} {Minasyan)}},
     journal = {Groups, geometry, and dynamics},
     pages = {1483--1515},
     year = {2023},
     volume = {17},
     number = {4},
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