1.1. VAN KAMPEN DIAGRAMS 3
1.1. Van Kampen diagrams
We recall some basic definitions and facts concerning Dehn functions and van
Kampen diagrams.
1.1.1. Dehn functions and isoperimetric inequalities. Given a finitely
presented group G = A | R and a word w in the generators A±1 that represents
1 G, one defines
Area(w) = min N
N+
| equality w =
N
j=1
uj
−1rjuj
in F (A) with rj
R±1
.
The Dehn function δ(n) of the finite presentation A | R is defined by
δ(n) = max{Area(w) | w ker(F (A) G), |w| n } ,
where |w| denotes the length of the word w. Whenever two presentations define
isomorphic (or indeed quasi-isometric) groups, the Dehn functions of the finite
presentations are equivalent under the relation that identifies functions [0, ∞)
[0, ∞) that only differ by a quasi-Lipschitz distortion of their domain and their
range.
For any constants p, q 1, one sees that n np is equivalent to n nq
only if p = q. Thus it makes sense to say that the “Dehn function of a group” is
np.
A group Γ is said to satisfy a quadratic isoperimetric inequality if its Dehn
function is n or
n2.
A result of Gromov [26], detailed proofs of which were
given by several authors, states that if a Dehn function is subquadratic, then it is
linear see [15, III.H] for a discussion, proof and references.
See [12] for a thorough and elementary account of what is known about Dehn
functions and an explanation of their connection with filling problems in Riemann-
ian geometry.
1.1.2. Van Kampen diagrams. According to van Kampen’s lemma (see
[27], [30] or [12]) an equality w =
N
j=1
ujrjuj
−1
in the free group A, with N =
Area(w), can be portrayed by a finite, 1-connected, combinatorial 2-complex with
basepoint, embedded in
R2.
Such a complex is called a van Kampen diagram for
w; its oriented 1-cells are labelled by elements of
A±1;
the boundary label on each
2-cell (read with clockwise orientation from one of its vertices) is an element of
R±1;
and the boundary cycle of the complex (read with positive orientation from
the basepoint) is the word w; the number of 2-cells in the diagram is N. Conversely,
any van Kampen diagram with M 2-cells gives rise to an equality in F (A) expressing
the word labelling the boundary cycle of the diagram as a product of M conjugates
of the defining relations. Thus Area(w) is the minimum number of 2-cells among
all van Kampen diagrams for w. If a van Kampen diagram for w has Area(w)
2-cells, then is a called a least-area diagram. If the underlying 2-complex is
homeomorphic to a 2-dimensional disc, then the van Kampen diagram is called a
disc diagram.
We use the term area to describe the number of 2-cells in a van Kampen dia-
gram, and write Area ∆. We write ∂∆ to denote the boundary cycle of the diagram;
we write |∂∆| to denote the length of this cycle.
Note that associated to a van Kampen diagram with basepoint p one has
a morphism of labelled, oriented graphs h∆ : (∆(1), p) (CA, 1), where CA is the
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