Van Kampen diagram

In the mathematical area of geometric group theory, a Van Kampen diagram (sometimes also called a Lyndon–Van Kampen diagram ) is a planar diagram used to represent the fact that a particular word in the generators of a group given by a group presentation represents the identity element in that group.
History.
The notion of a Van Kampen diagram was introduced by Egbert van Kampen in 1933. This paper appeared in the same issue of American Journal of Mathematics as another paper of Van Kampen, where he proved what is now known as the Seifert–Van Kampen theorem. The main result of the paper on Van Kampen diagrams, now known as the "van Kampen lemma" can be deduced from the Seifert–Van Kampen theorem by applying the latter to the presentation complex of a group. However, Van Kampen did not notice it at the time and this fact was only made explicit much later (see, e.g.). 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. Currently Van Kampen diagrams are a standard tool in geometric group theory. They are used, in particular, for the study of isoperimetric functions in groups, and their various generalizations such as isodiametric functions, filling length functions, and so on.
Formal definition.
The definitions and notations below largely follow Lyndon and Schupp.
Let 
be a group presentation where all "r"∈"R" are cyclically reduced words in the free group "F"("A"). 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 formula_2, given with a specific embedding formula_3 with the following additional data and satisfying the following additional properties:
Thus the 1-skeleton of formula_2 is a finite connected planar graph "Γ" embedded in formula_9 and the two-cells of formula_2 are precisely the bounded complementary regions for this graph.
By the choice of "R"∗ Condition 4 is equivalent to requiring that for each region of formula_2 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 formula_2 also has the "boundary cycle", denoted formula_13, which is an edge-path in the graph "Γ" corresponding to going around formula_2 once in the clockwise direction along the boundary of the unbounded complementary region of "Γ", starting and ending at the base-vertex of formula_2. The label of that boundary cycle is a word "w" in the alphabet "A" ∪ "A"−1 (which is not necessarily freely reduced) that is called the "boundary label" of formula_2.
Further terminology.
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:
Example.
The following figure shows an example of a Van Kampen diagram for the free abelian group of rank two
The boundary label of this diagram is the word
The area of this diagram is equal to 8.
Van Kampen lemma.
A key basic result in the theory is the so-called "Van Kampen lemma" which states the following:
Sketch of the proof.
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 formula_2. The inductive step consists in "peeling" off one of the boundary regions of formula_2 to get a Van Kampen diagram formula_36 with boundary cycle "w" and observing that in "F"("A") we have 
where "s"∈"R"∗ is the boundary cycle of the region that was removed to get formula_36 from formula_2.
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 formula_40 with boundary label "w"0 such that "w" = "w"0 in "F"("A") (after possibly freely reducing "w"0). Namely consider a representation of "w" of the form (♠) above. Then make formula_40 to be a wedge of "n" "lollipops" with "stems" labeled by "ui" and with the "candys" (2-cells) labelled by "si". Then the boundary label of formula_40 is a word "w"0 such that "w" = "w"0 in "F"("A"). However, it is possible that the word "w"0 is not freely reduced. One then starts performing "folding" moves to get a sequence of Van Kampen diagrams formula_43 by making their boundary labels more and more freely reduced and making sure that at each step the boundary label of each diagram in the sequence is equal to "w" in "F"("A"). The sequence terminates in a finite number of steps with a Van Kampen diagram formula_44 whose boundary label is freely reduced and thus equal to "w" as a word. The diagram formula_44 may not be reduced. 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 formula_2 whose boundary cycle is freely reduced and equal to "w".
Strengthened version of Van Kampen's lemma.
Moreover, the above proof shows that the conclusion of Van Kampen's lemma can be strengthened as follows. Part 1 can be strengthened to say that if formula_2 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".
Dehn functions and isoperimetric functions.
Area of a word representing the identity.
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).
One can show that the area of "w" can be equivalently defined as the smallest "n"≥0 such that there exists a representation (♠) expressing "w" as a product in "F"("A") of "n" conjugates of the defining relators.
Isoperimetric functions and Dehn functions.
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.
Then the "Dehn function" of (†) is defined as