[6] However, Van Kampen did not notice it at the time and this fact was only made explicit much later (see, e.g.[7]).
Van Kampen diagrams remained an underutilized tool in group theory for about thirty years, until the advent of the small cancellation theory in the 1960s, where Van Kampen diagrams play a central role.
[8] Currently Van Kampen diagrams are a standard tool in geometric group theory.
The alphabet A and the set of defining relations R are often assumed to be finite, which corresponds to a finite group presentation, but this assumption is not necessary for the general definition of a Van Kampen diagram.
Let R∗ be the symmetrized closure of R, that is, let R∗ be obtained from R by adding all cyclic permutations of elements of R and of their inverses.
A Van Kampen diagram over the presentation (†) is a planar finite cell complex
there is some boundary vertex of that region and some choice of direction (clockwise or counter-clockwise) such that the boundary label of the region read from that vertex and in that direction is freely reduced and belongs to R. A Van Kampen diagram
once in the clockwise direction along the boundary of the unbounded complementary region of Γ, starting and ending at the base-vertex of
In general, a Van Kampen diagram has a "cactus-like" structure where one or more disk-components joined by (possibly degenerate) arcs, see the figure below:
The following figure shows an example of a Van Kampen diagram for the free abelian group of rank two
A key basic result in the theory is the so-called Van Kampen lemma[9] which states the following: First observe that for an element w ∈ F(A) we have w = 1 in G if and only if w belongs to the normal closure of R in F(A) that is, if and only if w can be represented as where n ≥ 0 and where si ∈ R∗ for i = 1, ..., n. Part 1 of Van Kampen's lemma is proved by induction on the area of
The proof of part two of Van Kampen's lemma is more involved.
First, it is easy to see that if w is freely reduced and w = 1 in G there exists some Van Kampen diagram
One then starts performing "folding" moves to get a sequence of Van Kampen diagrams
The sequence terminates in a finite number of steps with a Van Kampen diagram
If that happens, we can remove the reduction pairs from this diagram by a simple surgery operation without affecting the boundary label.
Eventually this produces a reduced Van Kampen diagram
whose boundary cycle is freely reduced and equal to w. Moreover, the above proof shows that the conclusion of Van Kampen's lemma can be strengthened as follows.
is a Van Kampen diagram of area n with boundary label w then there exists a representation (♠) for w as a product in F(A) of exactly n conjugates of elements of R∗.
Part 2 can be strengthened to say that if w is freely reduced and admits a representation (♠) as a product in F(A) of n conjugates of elements of R∗ then there exists a reduced Van Kampen diagram with boundary label w and of area at most n. Let w ∈ F(A) be such that w = 1 in G. Then the area of w, denoted Area(w), is defined as the minimum of the areas of all Van Kampen diagrams with boundary labels w (Van Kampen's lemma says that at least one such diagram exists).
A nonnegative monotone nondecreasing function f(n) is said to be an isoperimetric function for presentation (†) if for every freely reduced word w such that w = 1 in G we have where |w| is the length of the word w. Suppose now that the alphabet A in (†) is finite.
Let w ∈ F(A) be a freely reduced word such that w = 1 in G. A Van Kampen diagram