Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra
(with Noel Brady, Martin Bridson, Krishnan Shankar)
Geometry and Topology 13 (2009), 141-187

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 Xr,P and determine the Dehn function of its fundamental group Gr,P in terms of r and the Perron-Frobenius eigenvalue of P. The range of functions obtained includes δ(x) = xs, where sQ ∩ [2, ∞) is arbitrary. Next, special features of the groups Gr,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 xs. Similar isoperimetric inequalities are obtained for fillings modeled on arbitrary manifold pairs (M, ∂M) in addition to (Bk+1, Sk).