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 nonnegative 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 delta(x) velence x~(s), where s E Q intersect [2, infinity) 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, (partial deriv)M) in addition to (B~(k+1), S~(k)).
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