Where does the idea of 'area' come from in Geometric Group Theory? The [wikipedia article](https://en.wikipedia.org/wiki/Dehn_function) states that this definition was 'inspired' from Riemannian geometry:



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> Gromov's proof was in large part informed by analogy with filling area functions for compact Riemannian manifolds where the area of a minimal surface bounding a null-homotopic closed curve is bounded in terms of the length of that curve
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Could someone explain what the relationship is between a geometric notion of area, and the formal definition [written down for completeness]:


Let $G = \langle S | R \rangle$ be a finitely presented group. Let $w$ be a word in the free group $F(S)$. Then if $w =\_G 1$, we can write:


$$
w = \prod\_{i=1}^n u\_i r\_i^{\pm 1} u\_i^{-1} \quad u\_i \in F(S); r\_i \in R
$$


The area of $w$ is defined as $\min \{ n : w = \prod\_{i=1}^n u\_i r\_i^{\pm 1}u\_i^{-1} \}$


It's intuitively obvious how the length of the perimeter can be related to $|w|$: if we think of the path walked by $w$ on the Cayley graph starting from (say) the identity, the path will be a loop (as $w =\_G 1$). The perimeter is the number of edges we need to traverse, which is the length of the word. On the other hand, this definition of area given above is not transparent. It seems to be saying something like:



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> count the minimum number of 'irreducible' components needed to write $w$ down.
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I am unable to see the geometric content of this definition. I would greatly appreciate one, either by analogy, or a direct explanation on the Cayley graph of $G$.



Geometric content of area of a word in geometric group theory?
Let $X$ be the presentation complex of $G=\langle S\mid R\rangle$. Any element $g\in G$ can be realised as a (based) loop $w:S^1\to X$, which we can take to be a cellular map.


A *van Kampen diagram* is a simply connected, planar 2-complex $D$ with a cellular map $D\to X$. An embedding of $D$ into the plane defines a natural boundary $\partial D$, namely the loop round the "outside" of $D$; $D$ is said to be a van Kampen diagram *for $w$* if the map $w:S^1\to X$ factors through the boundary of $D$.


The term *area* can now be very naturally motivated: the area of $w$ is just the minimum number of 2-cells in a van Kampen diagram for $w$. You can easily illustrate this definition by working out the areas of words in the presentation


$\langle a,b\mid [a,b]\rangle$


for $\mathbb{Z}^2$. For instance, the area of $[a^n,b^n]$ is $n^2$, but the area of $[a,b]a[a,b]a[a,b]\ldots a[a,b]a^{-n}$ is $n$, and both of these can be proved by drawing pictures in the plane.


As @YCor says in comments, this is all explained in Bridson's survey *The geometry of the word problem*.


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As requested, here's the definition of the *presentation complex* $X$ associated to the presentation $\langle S\mid R\rangle$.


Take a graph (ie a 1-dimensional CW-complex) $X^{(1)}$ with one vertex, and with edges in bijection with the elements of the generating set $S$. The fundamental group of this graph is naturally isomorphic to the free group $F(S)$, and so each relator $r\in R$ can be realised as a loop $\rho\_r$ in $X^{(1)}$. We now use the loops $\rho\_r$ as attaching maps for the 2-cells of $X$, and the Seifert--van Kampen theorem tells us that the fundamental group of $X$ is isomorphic to $G$. Note also that the Cayley graph $\mathrm{Cay}\_S(G)$ is naturally the 1-skeleton of the universal cover of $X$.


