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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
∈
ℚ
∩ [ 2 , ∞ )
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
≥ ( 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 ) 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
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