In the mathematical area of geometric group theory, a Van Kampen diagram (sometimes also called a Lyndon–Van Kampen diagram [1] [2] [3] ) 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.
The notion of a Van Kampen diagram was introduced by Egbert van Kampen in 1933. [4] 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. [5] 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. [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. 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.
The definitions and notations below largely follow Lyndon and Schupp. [9]
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 , given with a specific embedding with the following additional data and satisfying the following additional properties:
Thus the 1-skeleton of is a finite connected planar graph Γ embedded in and the two-cells of 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 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 also has the boundary cycle, denoted , which is an edge-path in the graph Γ corresponding to going around once in the clockwise direction along the boundary of the unbounded complementary region of Γ, starting and ending at the base-vertex of . 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 .
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
The boundary label of this diagram is the word
The area of this diagram is equal to 8.
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 inductive step consists in "peeling" off one of the boundary regions of to get a Van Kampen diagram 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 from .
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 with boundary label w0 such that w = w0 in F(A) (after possibly freely reducing w0). Namely consider a representation of w of the form (♠) above. Then make 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 is a word w0 such that w = w0 in F(A). However, it is possible that the word w0 is not freely reduced. One then starts performing "folding" moves to get a sequence of Van Kampen diagrams 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 whose boundary label is freely reduced and thus equal to w as a word. The diagram 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 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. [9] Part 1 can be strengthened to say that if 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).
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.
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
It is easy to see that Dehn(n) is an isoperimetric function for (†) and, moreover, if f(n) is any other isoperimetric function for (†) then Dehn(n) ≤ f(n) for every n ≥ 0.
Let w ∈ F(A) be a freely reduced word such that w = 1 in G. A Van Kampen diagram with boundary label w is called minimal if Minimal Van Kampen diagrams are discrete analogues of minimal surfaces in Riemannian geometry.
In the mathematical area of geometric group theory, a Van Kampen diagram (sometimes also called a Lyndon–Van Kampen diagram [1] [2] [3] ) 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.
The notion of a Van Kampen diagram was introduced by Egbert van Kampen in 1933. [4] 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. [5] 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. [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. 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.
The definitions and notations below largely follow Lyndon and Schupp. [9]
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 , given with a specific embedding with the following additional data and satisfying the following additional properties:
Thus the 1-skeleton of is a finite connected planar graph Γ embedded in and the two-cells of 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 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 also has the boundary cycle, denoted , which is an edge-path in the graph Γ corresponding to going around once in the clockwise direction along the boundary of the unbounded complementary region of Γ, starting and ending at the base-vertex of . 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 .
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
The boundary label of this diagram is the word
The area of this diagram is equal to 8.
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 inductive step consists in "peeling" off one of the boundary regions of to get a Van Kampen diagram 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 from .
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 with boundary label w0 such that w = w0 in F(A) (after possibly freely reducing w0). Namely consider a representation of w of the form (♠) above. Then make 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 is a word w0 such that w = w0 in F(A). However, it is possible that the word w0 is not freely reduced. One then starts performing "folding" moves to get a sequence of Van Kampen diagrams 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 whose boundary label is freely reduced and thus equal to w as a word. The diagram 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 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. [9] Part 1 can be strengthened to say that if 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).
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.
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
It is easy to see that Dehn(n) is an isoperimetric function for (†) and, moreover, if f(n) is any other isoperimetric function for (†) then Dehn(n) ≤ f(n) for every n ≥ 0.
Let w ∈ F(A) be a freely reduced word such that w = 1 in G. A Van Kampen diagram with boundary label w is called minimal if Minimal Van Kampen diagrams are discrete analogues of minimal surfaces in Riemannian geometry.