The Double exponential Theorem for isodiametric and isoperimetric Functions

The relevant definitions of terms occurring in the statement are as follows. If P = 〈x1, x2, . . . , xp | R1, R2, . . .Rq〉 is a finite presentation, we shall denote by G = G(P) the associated group; here G = F/N , where F is the free group freely generated by the generators x1, . . . , xp and N is the normal closure of the relators. If w is an element of F (which we may identify with a reduced word in the free basis), we write `(w) for the length of the word w and we write w =G 1 to mean that w represents the identity element of G. We shall make use of the terminology of van Kampen diagrams [L-S, p. 235ff]. We write AreaP (w) for the minimum number of faces (i.e 2-cells) in a van Kampen diagram with boundary label w. Equivalently, AreaP(w) is the minimum number of relators or inverses of relators occurring in all expressions of w as a product (in F ) of their conjugates. The function f : → is an isoperimetric function for P if, for all n and all words w with `(w) ≤ n and w =G 1, we have AreaP(w) ≤ f(n). The minimal isoperimetric function for P is called its Dehn function. If D is a van Kampen diagram with boundary label w, we choose the base point v0 in the boundary of D corresponding to where one starts reading the boundary label w and one defines