Homological and homotopical Dehn functions are different

@article{Abrams2013HomologicalAH,
  title={Homological and homotopical Dehn functions are different},
  author={Aaron Abrams and Noel Brady and Pallavi Dani and Robert Young},
  journal={Proceedings of the National Academy of Sciences},
  year={2013},
  volume={110},
  pages={19206 - 19212}
}
The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, whereas the homotopical Dehn function measures fillings of curves by disks. Because the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions; however, before this work, there were no known examples of… 
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References

SHOWING 1-10 OF 20 REFERENCES
Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra
The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and…
Pushing fillings in right‐angled Artin groups
TLDR
The results give sharp bounds on the higher Dehn func- tions of Bestvina-Brady groups, a complete characterization of the divergence of geodesics in RAAGs, and an upper bound for filling loops at infinity in the mapping class group.
The Geometry of the Word Problem for Finitely Generated Groups
Dehn Functions and Non-Positive Curvature.- The Isoperimetric Spectrum.- Dehn Functions of Subgroups of CAT(0) Groups.- Filling Functions.- Filling Functions.- Relationships Between Filling…
Dehn functions and finiteness properties of subgroups of perturbed right-angled Artin groups
We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction…
Metric Spaces of Non-Positive Curvature
This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by…
Homological and homotopical higher-order filling functions
We construct groups in which FV^3(n) != \delta^2(n). This construction also leads to groups G_k, k >= 3 for which \delta^{k}(n) is not subrecursive.
Finitely Presented Subgroups of Automatic Groups and their Isoperimetric Functions
We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of…
A SHORT PROOF OF GROMOV'S FILLING INEQUALITY
We give a very short and rather elementary proof of Gromov's filling volume inequality for n-dimensional Lipschitz cycles (with integer and Z 2 -coefficients) in L∞-spaces. This inequality is used in…
Morse theory and finiteness properties of groups
Abstract. We examine the finiteness properties of certain subgroups of “right angled” Artin groups. In particular, we find an example of a group that is of type FP(Z) but is not finitely presented.
An introduction to right-angled Artin groups
Recently, right-angled Artin groups have attracted much attention in geometric group theory. They have a rich structure of subgroups and nice algorithmic properties, and they give rise to cubical…
...
...