Equivalence of Geometric and Combinatorial Dehn Functions 3

We prove that if a finitely presented group acts properly discontinuously, cocompactly and by isometries on a simply connected Riemannian manifold, then the Dehn function of the group and the corresponding filling function of the manifold are equivalent, in a sense described below. 1. Dehn functions and their equivalence Let X be a simply connected 2-complex , and let w be an edge circuit in X. If D is a van Kampen diagram for w (see [5]), then the area of D is defined as the number of 2-cells on D, and the area of w, denoted a(w), is defined as the minimum of the areas of all van Kampen diagrams for w. The Dehn function of X is then defined to be δX(n) = max a(w), where the maximum is taken over all loops w of length l(w) ≤ n. Given two functions f and g from N to N (or, more generally, from R to R), we say that f ≺ g if there exist positive constants A, B, C, D, E so that f(n) ≤ Ag(Bn+ C) +Dn+ E. Two such functions are called equivalent (denoted f ≡ g) if f ≺ g and g ≺ f . The Dehn function is invariant under quasi-isometries: when one considers the 1skeleton of a complex as a metric space with the path metric, where every edge has length one, two complexes with quasi-isometric 1-skeleta have equivalent Dehn functions (see [1]). Let G be a finitely presented group, and let P be a finite presentation for G. Let K = K(P) be the 2-complex associated to P, i.e. the 2-complex with a single vertex, an oriented edge for every generator of P, and a 2-cell for every relator, attached to the edges according to the spelling of the relator. Then the Dehn function of P is, by definition, the Dehn function δK̃ of the universal covering of K. Two finite presentations P and Q for the same group G yield 2-complexes K̃(P) and K̃(Q) with quasi-isometric 1-skeleta, and hence equivalent Dehn functions. Thus the Dehn function of the group G is defined to be the equivalence class of the Dehn function of any of its presentations. An extensive treatment of Dehn functions of finitely presented groups is given in [4]. A closely related definition can be formulated in the context of Riemannian manifolds, dating back to the isoperimetric problem for R in the calculus of variations. Given a Lipschitz loop γ in a simply connected Riemannian manifold M , we define