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