Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra, Geom

ABSTRACT. 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 func-tion 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 posi-tive rational number r, we associate a finite, aspherical 2-complex Xr,P and calcu-late the Dehn function of its fundamental groupGr,P in terms of r and the Perron-Frobenius eigenvalue of P. The range of functions obtained includes δ(x) = xs, where s ∈ Q ∩ [2,∞) is arbitrary. By repeatedly forming multiple HNN exten-sions of the groups Gr,P we exhibit a similar range of behavior among higher-dimensional Dehn functions, proving in particular that for each positive integer k and rational s> (k + 1)/k there exists a group with k-dimensional Dehn func-tion xs. Similar isoperimetric inequalities are obtained for arbitrarymanifold pairs (M,∂M) in addition to (Bk+1, Sk)