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Abstract. We survey current work relating to isoperimetric functions and isodiametric functions of finite presentations. §1. Introduction and DefinitionsIsoperimetric functions are classical in differential geometry, but their use in group theory derives from Gromov's seminal article [Gr] and his characterization of word hyperbolic groups by a linear isoperimetric inequality. Isodiametric functions were introduced in our article [G1] in an attempt to provide a group theoretic framework for a result of Casson's (see Theorem 3.6 below). It turned out subsequently that the notion had been considered earlier under a different name [FHL]. We have learned since that the differential geometers also have their isodiametric functions and they mean something different by them. However the analogy is too suggestive to abandon this terminology and we shall retain it here. Up to an appropriate equivalence relation (Proposition 1.1 below), isoperimetric and isodiametric functions are quasiisometry invariants of finitely presented groups. Hence these functions are examples of geometric properties, in the terminology of [Gh].If P = x 1 , x 2 , . . . , x p | R 1 , R 2 , . . . R q 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 x 1 , . . . , x p 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 andw for the element of G represented by w. We shall use freely the terminology of van Kampen diagrams [LS, p. 235ff] in the sequel.We write Area P (w) for the minimum number of faces (i.e 2-cells) in a van Kampen diagram with boundary label w. Equivalently, Area P (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 andw = 1, we have Area P (w) ≤ f (n). The minimum such isoperimetric function is called the Dehn function of P.

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