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 u−1rjuj j 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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