The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each non-negative integer matrix P and positive rational number r, we associate a finite, aspherical 2-complex X_r,P and determine the Dehn function of its fundamental group G_r,P in terms of r and the Perron-Frobenius eigenvalue of P. The range of functions obtained includes δ(x) = x^s, where s ∈ Q ∩ [2, ∞) is arbitrary. Next, special features of the groups G_r,P allow us to construct iterated multiple HNN extensions which exhibit similar isoperimetric behavior in higher dimensions. In particular, for each positive integer k and rational s ≥ (k+1)/k, there exists a group with k-dimensional Dehn function x^s. Similar isoperimetric inequalities are obtained for fillings modeled on arbitrary manifold pairs (M, ∂M) in addition to (B^k+1, S^k).

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