Ytukyg

I am studying geometric group theory and metric spaces of bounded curvature. In particular, I was reading the seminal Gromov's Hyperbolic Groups and trying to filling the details. In the first page, Gromov introduce the following definition.

Definition: Let $Gamma$ be a finitely presented group and let $mathcal{P}=langle mathcal{A}mid mathcal{R}rangle$ be a finite presentation of $Gamma$. Consider the presentation complex $K_mathcal{P}$ associated to $mathcal{P}$, which has a unique $0$-cell, one labelled $1$-cell for each generator in $mathcal{A}$ and one $2$-cell for each relation in $mathcal{R}$ attached according to the relation. We can triangulate $mathbb{R}^5$ and $K_mathcal{P}$, and regard the later as a subcomplex of the former. Take a regular neighborhood $U$ of $K_mathcal{P}$ in $mathbb{R}^5$. We say that $Gamma$ is hyperbolic if $U$ satisfies a linear isoperimetric inequality.

There are several equivalent definitions of hyperbolic group. One of them says that a finitely generated group is hyperbolic if and only if it has a linear (equivalence class of) Dehn function. In order to prove the equivalence between the two definitions, I suppose that we have to take the boundary $V=partial U$ instead of $U$ in the Gromov's definition and apply the following theorem.

Filling Theorem: Let $M$ be a closed connected riemannian manifold and let $Gamma$ its fundamental group. The Dehn function $delta_Gamma$ and the isoperimetric function $text{Fill}_M$ are asymptotically equivalent.

A proof can be found in:

1. Martin R. Bridson, The geometry of the word problem. In invitations to geometry and topology (Martin R. Bridson and Simon M. Salamon, eds.), Oxford University Press, 2002.


2. José Burillo and Jennifer Taback, Equivalence of geometric and combinatorial Dehn functions, New York J. Math 8 (2002), 169–179.

This theorem is very interesting for me and I suppose that it has applications. But the two references above are the unique places in which I read about it.

My question is: Are there any interesting applications of the Filling
Theorem? An answer can be for example a (sketch of a) proof of a non-trivial result which uses the Filling Theorem or the reference of a textbook in which the Filling Theorem is used to prove several non-trivial results.

Regarding my background, I am confortable with the basics of riemannian geometry, algebraic and differential topology, and homological algebra. I am a beginner in metric geometry, geometric measure theory, geometric group theory and geometric topology so every remark or comment is also very welcome.

Thanks in advance!

Relevant definitions:

Two non-decreasing functions $f,g:mathbb{R}_{geq 0}to mathbb{R}_{geq 0}$ are asymptotically equivalent if there exists a positive real number $C$ such that $f(x)leq Cg(Cx+C)+Cx+C$ and $g(x)leq Cf(Cx+C)+Cx+C$ for every $x$ in $mathbb{R}_{geq 0}$. A non-decreasing function $h:mathbb{N}_0tomathbb{N}_0$ is asymptotically equivalent to a non-decreasing function $f:mathbb{R}_{geq 0}to mathbb{R}_{geq 0}$ if a non-decreasing extension $hat{h}:mathbb{R}_{geq 0}to mathbb{R}_{geq 0}$ of $h$ is asymptotically equivalent to $f$ in the previous sense.

The Dehn function asociated to a finite presentation $mathcal{P}=langle mathcal{A}mid mathcal{R}rangle$ is the minimum non-decreasing function $delta_mathcal{P}:mathbb{N}_0tomathbb{N}_0$ such that for every null-homotopic word $w$ in the alphabet $mathcal{A}$ there exists an equality
$$
w=prod_{i=1}^Nx_ir_i^{epsilon_i}x_i^{-1}
$$
in the free-group $F(mathcal{A})$, where $x_iin F(mathcal{A})$, $r_iin mathcal{R}$ and $epsilon_i in {1,-1}$ for every $i$, such that
$$
Nleq delta_mathcal{P}(|w|),
$$
where $|w|$ is the length of the word $w$. Dehn functions of presentations of the same group are asymptotically equivalent.

The filling function asociated to a riemannian manifold $M$ is the minimum non-decreasing function $text{Fill}_M:mathbb{R}_{geq 0} to mathbb{R}_{geq 0}$ such that for every null-homotopic Lipschitz loop $c:S^1to M$ and every positive real number $varepsilon$ there exists a Lipschitz filling disk $hat{c}:D^2to M$ of $c$ (I think that this means that $hat{c}|_{S^1}$ is a Lipschitz reparameterization of $c$) such that the parametric area of $hat{c}$ satisfies
$$
text{Area}(hat{c})leq text{Fill}(L(c))+varepsilon,
$$
where $L(c)$ is the length of $c$ and the parametric area is defined as
$$
text{Area}(hat{c})=int_{M}#hat{c}^{-1}(y)dmathcal{H}^2(y),
$$
where $mathcal{H}^2$ is the Hausdorff 2-dimensional measure on $M$, and $#hat{c}^{-1}(y)$ is the cardinality of the fiber of $y$.

EDIT:

Although I have selected an accepted answer, any future contribution is very welcome. In fact, I am working on the subject and maybe in the future I will post my own answer here with some interesting application.

For a specific application, if $M$ is a compact hyperbolic $n$-manifold then the Dehn function of $pi_1 M$ has a linear upper bound.

One need only prove the filling function has a linear upper bound, and for this it suffices to prove that the filling function of the universal covering space of $M$, which is the $n$-dimensional hyperbolic space $mathbb H^n$, has a linear upper bound. So, given any Lipschitz loop $c : S^1 to mathbb H^m$ one must construct the appropriate extension $hat c : D^2 to mathbb H^m$ (the notation $hat c$ simply means that it is an extension of $c$; no reparameterization is necessary). Choose any $p in mathbb H^m$ and define $hat c$ to be $p$ at the center of $D^2$ and to map any radius to a reparameterized geodesic. An easy integral gives the required bound to $text{Area}(hat c)$, based on bounds of triangle areas in $mathbb H^n$.