Van Kampen diagram
Encyclopedia
In the mathematical
area of geometric group theory
, a 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.
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.
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 boundary label of this diagram is the word
The area of this diagram is equal to 8.
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 2 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.
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.
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.
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 surface
s in Riemannian geometry
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
area of geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
, a van Kampen diagram is a planar diagram used to represent the fact that a particular word in the generators
Generating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
given by a group presentation represents the identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
in that group.
History
The notion of a van Kampen diagram was introduced by Egbert van KampenEgbert van Kampen
Egbert Rudolf van Kampen was a mathematician. He made important contributions to topology, especially to the study of fundamental groups....
in 1933. This paper appeared in the same issue of American Journal of Mathematics
American Journal of Mathematics
The American Journal of Mathematics is a bimonthly mathematics journal published by the Johns Hopkins University Press.- History :The American Journal of Mathematics is the oldest continuously-published mathematical journal in the United States, established in 1878 at the Johns Hopkins University...
as another paper of van Kampen, where he proved what is now known as the Seifert–van Kampen theorem
Seifert–van Kampen theorem
In mathematics, the Seifert-van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X...
. 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
Seifert–van Kampen theorem
In mathematics, the Seifert-van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X...
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
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
for about thirty years, until the advent of the small cancellation theory
Small cancellation theory
In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "small overlaps" with each other. It turns out that small cancellation conditions have substantial...
in the 1960s, where van Kampen diagrams play a central role. Currently van Kampen diagrams are a standard tool in geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
. 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&Schupp.Let
(†)be a group presentation where all r∈R are cyclically reduced words in the free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
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:- The complex
is connected and simply connected. - Each edge (one-cell) of
is labelled by an arrow and a letter a∈A. - Some vertex (zero-cell) which belongs to the topological boundary of
is specified as a base-vertex. - For each region (two-cell) of
for every vertex the boundary cycle of that region and for each of the two choices of direction (clockwise or counter-clockwise) the label of the boundary cycle of the region read from that vertex and in that direction is a freely reduced word in F(A) that belongs to R∗.
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 .Further terminology
- A van Kampen diagram
is called a disk diagram if 
is a topological disk, that is, when every edge of 
is a boundary edge of some region of 
and when 
has no cut-vertices. - A van Kampen diagram
is called non-reduced if there exists a reduction pair in 
, that is a pair of distinct regions of 
such that their boundary cycles share a common edge and such that their boundary cycles, read starting from that edge, clockwise for one of the regions and counter-clockwise for the other, are equal as words in A ∪ A−1. If no such pair of region exists, 
is called reduced. - The number of regions (two-cells) of
is called the area of 
denoted 
.
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 twoThe 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:- Let
be a van Kampen diagram over the presentation (†) with boundary label w which is a word (not necessarily freely reduced) in the alphabet A ∪ A−1. Then w=1 in G. - Let w be a freely reduced word in the alphabet A ∪ A−1 such that w=1 in G. Then there exists a reduced van Kampen diagram
over the presentation (†) whose boundary label is freely reduced and is equal to w.
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 closureNormal closure
The term normal closure is used in two senses in mathematics:* In group theory, the normal closure of a subset of a group is the smallest normal subgroup that contains the subset; see conjugate 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 havewhere s∈R∗ is the boundary cycle of the region that was removed to get
from .The proof of part 2 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.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 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.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 havewhere |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 surfaceMinimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
s in Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
.
Generalizations and other applications
- There are several generalizations of van-Kampen diagrams where instead of being planar, connected and simply connected (which means being homotopically equivalent to a disk) the diagram is drawn on or homotopically equivalent to some other surface. It turns out, that there is a close connection between the geometry of the surface and certain group theoretical notions. A particularly important one of these is the notion of an annular van Kampen diagram, which is homotopically equivalent to an annulusAnnulus (mathematics)In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...
. Annular diagrams, also known as conjugacy diagrams, can be used to represent conjugacyConjugacy classIn mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
in groups given by group presentations. Also spherical van Kampen diagrams are related to several versions of group-theoretic asphericityAspherical spaceIn topology, a branch of mathematics, an aspherical space is a topological space with all higher homotopy groups equal to 0.If one works with CW complexes, one can reformulate this condition: an aspherical CW complex is a CW complex whose universal cover is contractible. Indeed, contractibility of...
and to Whitehead's asphericity conjectureWhitehead conjectureThe Whitehead conjecture is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcomplex of a two-dimensional aspherical CW complex is aspherical....
, Van Kampen diagrams on the torus are related to commuting elements, diagrams on the real projective plane are related to involutions in the group and diagrams on Klein's bottle are related to elements that are conjugated to their own inverse.
- Van Kampen diagrams are central objects in the small cancellation theorySmall cancellation theoryIn the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "small overlaps" with each other. It turns out that small cancellation conditions have substantial...
developed by Greendlinger, Lyndon and Schupp in the 1960s-1970s. Small cancellation theory deals with group presentations where the defining relations have "small overlaps" with each other. This condition is reflected in the geometry of reduced van Kampen diagrams over small cancellation presentations, forcing certain kinds of non-positively curved or negatively cn curved behavior. This behavior yields useful information about algebraic and algorithmic properties of small cancellation groups, in particular regarding the word and the conjugacy problems. Small cancellation theory was one of the key precursors of geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
, that emerged as a distinct mathematical area in lated 1980s and it remains an important part of geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
.
- Van Kampen diagrams play a key role in the theory of word-hyperbolic groups introduced by Gromov in 1987. In particular, it turns out that a finitely presented group is word-hyperbolic if and only if it satisfies a linear isoperimetric inequality. Moreover, there is an isoperimetric gap in the possible spectrum of isomperimetric functions for finitely presented groups: for any finitely presented group either it is hyperbolic and satisfies a linear isoperimetric inequality or else the Dehn function is at least quadratic.
- The study of isoperimetric functions for finitely presented groups has become an important general theme in geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
where substantial progress has occurred. Much work has gone into constructing groups with "fractional" Dehn functions (that is, with Dehn functions being polynomials of non-integer degree). The work of RipsEliyahu RipsEliyahu Rips, also Ilya Rips is a Latvian-born Israeli mathematician known for his research in geometric group theory. He became known to the general public following his coauthoring a paper on the Torah Code....
, Ol'shanskii, Birget and Sapir explored the connections between Dehn functions and time complexity functions of Turing machineTuring machineA Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
s and showed that an arbitrary "reasonable" time function can be realized (up to appropriate equivalence) as the Dehn function of some finitely presented group.
- Various stratified and relativized versions of van Kampen diagrams have been explored in the subject as well. In particular, a stratified version of small cancellation theory, developed by Ol'shanskii, resulted in constructions of various group-theoretic "monsters", such as the Tarski Monster, and in geometric solutions of the Burnside problem for periodic groups of large exponent. Relative versions of van Kampen diagrams (with respect to a collection of subgroups) were used by Osin to develop an isoperimetric function approach to the theory of relatively hyperbolic groupRelatively hyperbolic groupIn mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group...
s.
See also
- Geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
- Presentation of a groupPresentation of a groupIn mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
- Seifert–van Kampen theoremSeifert–van Kampen theoremIn mathematics, the Seifert-van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X...
Basic references
- Alexander Yu. Ol'shanskii. Geometry of defining relations in groups. Translated from the 1989 Russian original by Yu. A. Bakhturin. Mathematics and its Applications (Soviet Series), 70. Kluwer Academic Publishers Group, Dordrecht, 1991. ISBN 0-7923-1394-1
- Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. ISBN 9783540411581; Ch. V. Small Cancellation Theory. pp. 235–294.