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Where does the idea of 'area' come from in Geometric Group Theory? The wikipedia article states that this definition was 'inspired' from Riemannian geometry:

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

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:

count the minimum number of 'irreducible' components needed to write $w$ down.

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 $G$.

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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 $X$.

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  • $\begingroup$ Thank you. I am unaware of this notion of 'presentation complex'. I presume the survey The geometry of the word problem. spells it out in full detail. It would still perhaps be nice if you could append its definition to the answer so the answer is self-contained. $\endgroup$ – Siddharth Bhat Jun 18 at 17:18
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    $\begingroup$ I've added the definition to the answer. $\endgroup$ – HJRW Jun 18 at 17:26
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    $\begingroup$ If you want a concrete example of a presentation complex, start with the standard presentation $\pi_1(S) = \langle a_1, b_1 ,a_2, b_2 \mid a_1 b_1 a_1^{-1} b_1^{-1} a_2 b_2 a_2 b_2^{-1}\rangle$ of the fundamental group of a closed oriented surface $S$ of genus $2$. $\endgroup$ – Lee Mosher Jun 27 at 14:41
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    $\begingroup$ Use this presentation to define a gluing diagram on a regular octagon with angles $2\pi/8$ in the hyperblic plane $\mathbb H^2$. That octagon is a fundamental domain for the deck transformation action of $\pi_1(S)$ on $\mathbb H^2$, and the translates of that octagon under the action define a tiling of $\mathbb H^2$. That tiling is the presentation complex for the standard presentation of $\pi_1(S)$. $\endgroup$ – Lee Mosher Jun 27 at 14:41
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    $\begingroup$ This also gives a concrete example of the relation between the geometric group theory notion of area and true area in the hyperbolic plane, where the latter is scaled so that the octagon has area 1. $\endgroup$ – Lee Mosher Jun 27 at 14:42

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