Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra

Brady, Noel; Bridson, Martin R; Forester, Max; Shankar, Krishnan

doi:10.2140/gt.2009.13.141

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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

DOI: 10.2140/gt.2009.13.141

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 $δ (x) = x^{s}$ , where $s \in ℚ \cap [2, \infty)$ 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 $s \geq (k + 1) ∕ k$ , there exists a group with $k$ –dimensional Dehn function $x^{s}$ . Similar isoperimetric inequalities are obtained for fillings modeled on arbitrary manifold pairs $(M, \partial M)$ in addition to $(B^{k + 1}, S^{k})$ .

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

References

Publication

Received: 23 November 2006

Revised: 17 September 2008

Accepted: 19 August 2008

Preview posted: 22 October 2008

Published: 31 December 2008

Proposed: Walter Neumann

Seconded: Benson Farb, Cameron Gordon

Authors

Noel Brady
Mathematics Department University of Oklahoma Norman, OK 73019 USA

Martin R Bridson
Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB UK

Max Forester
Mathematics Department University of Oklahoma Norman, OK 73019 USA

Krishnan Shankar
Mathematics Department University of Oklahoma Norman, OK 73019 USA

Volume 13, issue 1 (2009)