#### Volume 13, issue 1 (2009)

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Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra

### Noel Brady, Martin R Bridson, Max Forester and Krishnan Shankar

Geometry & Topology 13 (2009) 141–187
##### 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 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 $\delta \left(x\right)={x}^{s}$, where $s\in ℚ\cap \left[2,\infty \right)$ 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 $s\ge \left(k+1\right)∕k$, there exists a group with $k$–dimensional Dehn function ${x}^{s}$. Similar isoperimetric inequalities are obtained for fillings modeled on arbitrary manifold pairs $\left(M,\partial M\right)$ in addition to $\left({B}^{k+1},{S}^{k}\right)$.

##### Keywords
Dehn function, isoperimetric inequality, filling invariant, isoperimetric spectrum, high dimensional Dehn function, subgroup distortion
##### Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 20F69, 20E06, 57M07, 57M20, 53C99