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:
- 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.
- 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 $h:\mathbb{N}_0\to\mathbb{N}_0$ 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 $\delta_\mathcal{P}:\mathbb{N}_0\to\mathbb{N}_0$ 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 $x_i\in F(\mathcal{A})$, $r_i\in \mathcal{R}$ and $\epsilon_i \in \{1,-1\}$ for every $i$, such that
$$
N\leq \delta_\mathcal{P}(|w|),
$$
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 $c:S^1\to M$ and every positive real number $\varepsilon$ there exists a Lipschitz filling disk $\hat{c}:D^2\to M$ 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
$$
\text{Area}(\hat{c})=\int_{M}\#\hat{c}^{-1}(y)d\mathcal{H}^2(y),
$$
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.
Best Answer
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 $\mathbb H^n$.