**Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric
spectra**

(with
Noel Brady,
Martin Bridson,
Krishnan Shankar)

Geometry and 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 non-negative 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}).