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
| equality w =
in F (A) with rj
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
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
A group Γ is said to satisfy a quadratic isoperimetric inequality if its Dehn
function is n or
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 =
in the free group A, with N =
Area(w), can be portrayed by a finite, 1-connected, combinatorial 2-complex with
basepoint, embedded in
Such a complex is called a van Kampen diagram for
w; its oriented 1-cells are labelled by elements of
the boundary label on each
2-cell (read with clockwise orientation from one of its vertices) is an element of
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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