Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra

Brady, Noel¹ ; Bridson, Martin R² ; Forester, Max¹ ; Shankar, Krishnan¹

¹ Mathematics Department, University of Oklahoma, Norman, OK 73019, USA
² Mathematical Institute, 24-29 St Giles’, Oxford, OX1 3LB, UK

Geometry & topology, Tome 13 (2009) no. 1, pp. 141-187.

Résumé

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 $s\in \mathbb{Q}\cap \left[2,\infty \right)$ 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 $\left(B^{k+1},S^k\right)$.

Détail

DOI : 10.2140/gt.2009.13.141

Keywords: Dehn function, isoperimetric inequality, filling invariant, isoperimetric spectrum, high dimensional Dehn function, subgroup distortion

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