Abstract
The –dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of –spheres mapped into –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 and positive rational number , we associate a finite, aspherical –complex and determine the Dehn function of its fundamental group in terms of and the Perron–Frobenius eigenvalue of . The range of functions obtained includes , where is arbitrary. Next, special features of the groups allow us to construct iterated multiple HNN extensions which exhibit similar isoperimetric behavior in higher dimensions. In particular, for each positive integer and rational , there exists a group with –dimensional Dehn function . Similar isoperimetric inequalities are obtained for fillings modeled on arbitrary manifold pairs in addition to .
Citation
Noel Brady. Martin R Bridson. Max Forester. Krishnan Shankar. "Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra." Geom. Topol. 13 (1) 141 - 187, 2009. https://doi.org/10.2140/gt.2009.13.141
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