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 is arbitrary rational number at least 2. 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 ≥ (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,∂M) 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

Primary: 20F65

Secondary: 20E06, 20F69, 53C99, 57M07, 57M20

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