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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

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 Xr,P and determine the Dehn function of its fundamental group Gr,P in terms of r and the Perron–Frobenius eigenvalue of P. The range of functions obtained includes δ(x) = xs, where s [2,) is arbitrary. Next, special features of the groups Gr,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 xs. Similar isoperimetric inequalities are obtained for fillings modeled on arbitrary manifold pairs (M,M) in addition to (Bk+1,Sk).

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
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
Noel Brady
Mathematics Department
University of Oklahoma
Norman, OK 73019
Martin R Bridson
Mathematical Institute
24-29 St Giles’
Max Forester
Mathematics Department
University of Oklahoma
Norman, OK 73019
Krishnan Shankar
Mathematics Department
University of Oklahoma
Norman, OK 73019