# 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…

## 14 Citations

Homological Dehn functions of groups of type $FP_2$.

- Mathematics
- 2020

We prove foundational results for homological Dehn functions of groups of type $FP_2$ such as superadditivity and the invariance under quasi-isometry. We then study the homological Dehn functions of…

Pushing fillings in right‐angled Artin groups

- MathematicsJ. Lond. Math. Soc.
- 2013

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.

Combinatorial higher dimensional isoperimetry and divergence

- MathematicsJournal of Topology and Analysis
- 2019

In this paper we provide a framework for the study of isoperimetric problems in finitely generated groups, through a combinatorial study of universal covers of compact simplicial complexes. We show…

Subgroups of relatively hyperbolic groups of Bredon cohomological dimension 2

- Mathematics
- 2015

Abstract A remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension 2 is closed under taking finitely presented (or more generally FP 2 {\mathrm{FP}_{2}} )…

Geometric presentations of Lie groups and their Dehn functions

- Mathematics
- 2017

We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie…

Homological Filling Functions with Coefficients

- Mathematics
- 2020

How hard is it to fill a loop in a Cayley graph with an unoriented surface? Following a comment of Gromov in "Asymptotic invariants of infinite groups", we define homological filling functions of…

Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups

- Mathematics
- 2019

This article is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We…

Linear isoperimetric functions for surfaces in hyperbolic groups

- Mathematics
- 2022

. We show that word-hyperbolic groups satisfy linear isoperimetric functions for all homotopy types of surface diagrams. This generalises the linear isoperimetric functions for disc and annular…

Snowflake geometry in CAT (0) groups

- Mathematics
- 2016

We construct CAT (0) groups containing subgroups whose Dehn functions are given by xs , for a dense set of numbers s∈[2,∞) . This significantly expands the known geometric behavior of subgroups of…

Finiteness of Homological Filling Functions

- Mathematics
- 2016

Let $G$ be a group. For any $\mathbb{Z} G$--module $M$ and any integer $d>0$, we define a function $FV_{M}^{d+1}\colon \mathbb{N} \to \mathbb{N} \cup \{\infty\}$ generalizing the notion of…

## References

SHOWING 1-10 OF 20 REFERENCES

Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra

- Mathematics
- 2006

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

- MathematicsJ. Lond. Math. Soc.
- 2013

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

- Mathematics
- 2007

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

- Mathematics
- 2011

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

- Mathematics
- 1999

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

- Mathematics
- 2008

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

- Mathematics
- 1996

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

- Mathematics
- 2007

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

- Mathematics
- 1997

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

- Mathematics
- 2006

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…