Van Kampen diagram
[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.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 complexthere 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 diagramonce in the clockwise direction along the boundary of the unbounded complementary region of Γ, starting and ending at the base-vertex ofIn 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 twoA 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 ofFirst, it is easy to see that if w is freely reduced and w = 1 in G there exists some Van Kampen diagramOne then starts performing "folding" moves to get a sequence of Van Kampen diagramsThe sequence terminates in a finite number of steps with a Van Kampen diagramIf that happens, we can remove the reduction pairs from this diagram by a simple surgery operation without affecting the boundary label.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∗.