Van Kampen diagram

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 among 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. [E. van Kampen. [http://www.jstor.org/pss/2371129 "On some lemmas in the theory of groups".] American Journal of Mathematics.vol. 55, (1933), pp. 268–273] 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. [E. R. van Kampen. "On the connection between the fundamental groups of some related spaces". American Journal of Mathematics, vol. 55 (1933), pp. 261–267] 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. 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] ). 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. [Bruce Chandler, and Wilhelm Magnus. "The history of combinatorial group theory. A case study in the history of ideas." Studies in the History of Mathematics and Physical Sciences, 9. Springer-Verlag, New York, 1982. ISBN: 0-387-90749-1] 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&Schupp.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-13: 9783540411581; Ch. V. Small Cancellation Theory. pp. 235–294.]

Let :$G=langle\; A\; |\; R,\; angle$ (†)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 $mathcal\; D,$, given with a specific embedding $mathcal\; Dsubseteq\; mathbb\; R^2,$ with the following additional data and satisfying the following additional properties:

#The complex $mathcal\; D,$ is connected and simply connected.
#Each "edge" (one-cell) of $mathcal\; D,$ is labelled by an arrow and a letter "a"∈"A".
#Some "vertex" (zero-cell) which belongs to the topological boundary of $mathcal\; Dsubseteq\; mathbb\; R^2,$ is specified as a "base-vertex".
#For each "region" (two-cell) of $mathcal\; D,$ 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 $mathcal\; D,$ is a finite connected planar graph "Γ" embedded in $mathbb\; R^2,$ and the two-cells of $mathcal\; D,$ 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 $mathcal\; D,$ 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 $mathcal\; D,$ also has the "boundary cycle", denoted $partialmathcal\; D,$, which is an edge-path in the graph "Γ" corresponding to going around $mathcal\; D,$ once in the clockwise direction along the boundary of the unbounded complementary region of "Γ", starting and ending at the base-vertex of $mathcal\; D,$. 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 $mathcal\; D,$.

Further terminology

*A van Kampen diagram $mathcal\; D,$ is called a "disk diagram" if $mathcal\; D,$ is a topological disk, that is, when every edge of $mathcal\; D,$ is a boundary edge of some region of $mathcal\; D,$ and when $mathcal\; D,$ has no cut-vertices.
*A van Kampen diagram $mathcal\; D,$ is called "non-reduced" if there exists a "reduction pair" in $mathcal\; D,$, that is a pair of distinct regions of $mathcal\; D,$ such that their boundary cycles share a common edge and such that their bounadry 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, $mathcal\; D,$ is called "reduced".
*The number of regions (two-cells) of $mathcal\; D,$ is called the "area" of $mathcal\; D,$ denoted $\{\; m\; Area\}(mathcal\; D),$.

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:$G=langle\; a,\; b|\; ab\; a^\{-1\}b^\{-1\}\; angle.$

The boundary label of this diagram is the word:$w=b^\{-1\}b^3a^\{-1\}b^\{-2\}ab^\{-1\}ba^\{-1\}ab^\{-1\}ba^\{-1\}a.$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:

#Let $mathcal\; D,$ 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 $mathcal\; D,$ over the presentation (†) whose boundary label is freely reduced and is equal to "w".

ketch 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

:$w=u\_1s\_1u\_1^\{-1\}cdots\; u\_n\; s\_nu\_\{n\}^\{-1\}\; ext\{\; in\; \}\; F(A),$ (♠)

where "n" ≥ 0 and where "s_i" ∈ "R"_∗ for "i" = 1, ..., "n".

Part 1 of van Kampen's lemma is proved by induction on the area of $mathcal\; D,$. The unductive step consists in "peeling" off one of the boundary reagions of $mathcal\; D,$ to get a van Kampen diagram $mathcal\; D\text{'},$ with boundary cycle "w"' and observing that in "F"("A") we have :$w=usu^\{-1\}\; w\text{'},,$where "s"∈"R"_∗ is the boundary cycle of the region that was removed to get $mathcal\; D\text{'},$ from $mathcal\; D,$.

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 $mathcal\; D\_0,$ 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 $mathcal\; D\_0,$ to be a wedge of "n" "lollipops" with "stems" labeled by "u_i" and with the "candys" (2-cells) labelled by "s_i". Then the boundary label of $mathcal\; D\_0,$ 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 $mathcal\; D\_0,\; mathcal\; D\_1,\; mathcal\; D\_2,dots,$ 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 $mathcal\; D\_k,$ whose boundary label is freely reduced and thus equal to "w" as a word. The diagram $mathcal\; D\_k,$ 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 $mathcal\; D,$ whose boundary cycle is freely reduced and equal to "w".

trengthened 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 $mathcal\; D,$ 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

:$\{\; m\; Area\}(w)le\; f(|w|),$

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

:$\{\; m\; Dehn\}(n)=max$ m Area}(w): w=1 ext{ in } G, |w|le n, w ext{ freely reduced}.}

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 $mathcal\; D,$ with boundary label "w" is called "minimal" if $\{\; m\; Area\}(mathcal\; D)=\{\; m\; Area\}(w).$ Minimal van Kampen diagrams are discrete analogues of minimal surfaces in Riemannian geometry.

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. A particularly important one of these is the notion of an "annular van Kampen diagram", which is homotopically equivalent to an annulus. Annular diagrams, also known as "conjugacy diagrams", can be used to represent conjugacy in groups given by group presentations. Also "spherical van Kampen diagrams" are related to several versions of group-theoretic asphericity and to Whitehead's asphericity conjecture. [Ian M. Chiswell, Donald J. Collins, and Johannes Huebschmann. [http://www.springerlink.com/content/rx371t3406887116/ "Aspherical group presentations."] Mathematische Zeitschrift, vol. 178 (1981), no. 1, pp. 1–36.]

*Van Kampen diagrams are central objects in the small cancellation theory developed by Greendlinger, Lyndon and Schupp in the 1960s-1970s. [Martin Greendlinger. [http://www3.interscience.wiley.com/journal/113397463/abstract?CRETRY=1&SRETRY=0 "Dehn's algorithm for the word problem."] Communications in Pure and Applied Mathematics, vol. 13 (1960), pp. 67–83 ] 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 theory, that emerged as a distinct mathematical area in lated 1980s and it remains an important part of geometric group theory.

*Van Kampen diagrams play a key role in the theory of word-hyperbolic groups introduced by Gromov in 1987. [M. Gromov. "Hyperbolic Groups". Essays in Group Theory (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263; ISBN: 0-387-96618-8] 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. [Michel Coornaert, Thomas Delzant, Athanase Papadopoulos, "Géométrie et théorie des groupes: les groupes hyperboliques de Gromov". Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990. ISBN 3-540-52977-2] [B. H. Bowditch. "A short proof that a subquadratic isoperimetric inequality implies a linear one." Michigan Mathematical Journal, vol. 42 (1995), no. 1, pp. 103–107]

*The study of isoperimetric functions for finitely presented groups has become an important general theme in geometric group theory 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). [M. R. Bridson, [http://www.ams.org/jams/1999-12-04/S0894-0347-99-00308-2/home.html "Fractional isoperimetric inequalities and subgroup distortion."] Journal of the American Mathematical Society, vol. 12 (1999), no. 4, pp. 1103–1118 ] The work of Rips, Ol'shanskii, Birget and Sapir [M. Sapir, J.-C. Birget, E. Rips, "Isoperimetric and isodiametric functions of groups." Annals of Mathematics (2), vol. 156 (2002), no. 2, pp. 345–466] [J.-C. Birget, A. Yu. Ol'shanskii, E. Rips, M. Sapir, "Isoperimetric functions of groups and computational complexity of the word problem." Annals of Mathematics (2), vol. 156 (2002), no. 2, pp. 467–518] explored the connections between Dehn functions and time complexity functions of Turing machines 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 [A. Yu. Ol'sanskii."Infinite groups with cyclic subgroups." (in Russian) Doklady Akademii Nauk SSSR 245 (1979), no. 4, pp. 785–787. ] , and in geometric solutions of the Burnside problem for periodic groups of large exponent. [A. Yu. Ol'shanskii."On a geometric method in the combinatorial group theory." Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pp. 415–424, PWN, Warsaw, 1984] [S. V. Ivanov. "The free Burnside groups of sufficiently large exponents." International Journal of Algebra and Computation, vol. 4 (1994), no. 1-2.] Relative versions of van Kampen diagrams (with respect to a collections of subgroup) were used by Osin to develop an isoperimetric function approach to the theory of relatively hyperbolic groups. [Denis V. Osin. "Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems." Memoirs of the American Mathematical Society 179 (2006), no. 843 ]

ee also

*Geometric group theory
*Presentation of a group
*Seifert–van Kampen theorem

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-13: 9783540411581; Ch. V. Small Cancellation Theory. pp. 235–294.

Footnotes

External links

* [http://www.maths.bris.ac.uk/~matrr/Art/diagrams.pdf A collection of diagrams by Tim Riley]
* [http://euler.slu.edu/Dept/Faculty/jackson/research/diagrams.html van Kampen diagrams from the files of David A. Jackson]