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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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Let X be a topological space.IN [2] it was shown how one could define "generalized sheaf cohomology" of X; this generalizes ordinary sheaf cohomology, as well as generalized cohomology in the sense of [Ii]. In case X = Spec A, where A is a regular commutative ring, it was announced in [5] that the (Karoubi-Villamayor)K-groups of A could be obtained as generalized sheaf cohomology groups of X. As a consequence, one obtains a "local to global"(or "Atiyah-Hirzebruch") spectral sequence E~q= KP(×,~_q)~ K_(p+q)(A),where ~ -q is the (abelian) sheaf of local K-groups of A.[Note:In the Karoubi-Villamayor notation, one would write ~q and K p+q instead of ~ -q and K_(p+q).]This application of the results of [2] to K-theory requires an improvement of those results. In particular, one needs to eliminate a boundedness assumption on the coefficient sheaves for generalized sheaf cohomology; this can be done provided X is a Noetherian space of finite Krull dimension. It is the purpose of the present paper to present this improvement and to give the application to K-theory.Since the improvement vastly simplifies the generalized sheaf cohomology theory, we give in Sections i and 2 an account of that theory independent of [2].

If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are defined for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbolicity. A sufficient condition for hyperbolicity of a general graph is given along with an application to ‘relative hyperbolicity’. Finitely presented subgroups of Lyndon's small cancellation groups of hyperbolic type are word hyperbolic. Finitely presented subgroups of hyperbolic 1‐relator groups are hyperbolic. Finitely presented subgroups of free Burnside groups are finite in the stable range.

A finite CAT(0) 2-complex X is produced whose universal cover possesses two geodesic rays which diverge quadratically and such that no pair of rays diverges faster than quadratically. This example shows that an aphorism in Riemannian geometry, that predicts that in nonpositive curvature nonasymptotic geodesic rays either diverge exponentially or diverge linearly, does not hold in the setting of CAT (0) complexes. The fundamental group of X is that of a compact Riemannian manifold with totally geodesic boundary and nonpositive sectional curvature.