Snowflake groups, Perron-Frobenius eigenvalues and isoperimetric spectra - Brady, N; Bridson, M; Forester, M; Shankar, K (2009)
The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ballfillings of k-spheres mapped into kconnected 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 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 s is arbitrary rational number at least 2. 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) addition to (Bk+1,Sk). © 2009 Mathematical Sciences Publishers.