No 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 ı.x/ D x s , where s 2 Q \ OE2; 1/ 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 C 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 kC1 ; S k /.20F65; 20F69, 20E06, 57M07, 57M20, 53C99 IntroductionGiven a k -connected complex or manifold one wants to identify functions that bound the volume of efficient ball-fillings for spheres mapped into that space. The purpose of this article is to advance the understanding of which functions can arise when one seeks optimal bounds in the universal cover of a compact space. Despite the geometric nature of both the problem and its solutions, our initial impetus for studying isoperimetric problems comes from algebra, more specifically the word problem for groups.The quest to understand the complexity of word problems has been at the heart of combinatorial group theory since its inception. When one attacks the word problem for a finitely presented group G directly, the most natural measure of complexity is What Brady and Bridson actually do in [3] is associate to each pair of positive integers p > q a finite aspherical 2-complex whose fundamental group G p;q has Dehn function x 2 log 2 2p=q . These complexes are obtained by attaching a pair of annuli to a torus, the attaching maps being chosen so as to ensure the existence of a family of discs in the universal cover that display a certain snowflake geometry (cf Figure 4 below). In the present article we present a more sophisticated version of the snowflake construction that yields a much larger class of isoperimetric exponents.Theorem A Let P be an irreducible nonnegative integer matrix with Perron-Frobenius eigenvalue > 1, and let r be a rational number greater than every row sum of P . Then there is a finitely presented group G r;P with Dehn function ı.x/ ' x 2 log .r / .Here, ' denotes coarse Lipschitz equivalence of functions. By taking P to be the 1 1 matrix .2 2q / and r D 2 p (for integers p > 2q ) we obtain the Dehn function ı.x/ ' x p=q and deduce the following corollary.Corollary B Q \ .2; 1/ IP. For each positive integer k one has the k -dimensional isoperimetric spe...

ABy considering branched coverings of piecewise Euclidean cubical complexes, the paper provides an example of a torsion free hyperbolic group containing a finitely presented subgroup which is not hyperbolic.

The closure of the set of isoperimetric exponents for finitely presented groups is {1} ∪ [2, ∞). For each pair of positive integers p ≥ q, one can construct groups with aspherical presentations for which the Dehn function is n 2α , where α = log 2 (2p/q).Understanding the complexity of word problems in groups has been one of the central themes of combinatorial group theory in the twentieth century since the pioneering work of Max Dehn. In recent years, following the influential work of Mikhael Gromov [Gr1], attention has focused on estimating the complexity of the word problem by means of isoperimetric inequalities. This approach is based on the close connection between word problems in finitely presented groups and Plateau's problem concerning the filling of loops in Riemannian manifolds.A natural approach to the filling problem for loops in the universal cover of a closed Riemannian manifold M is to seek isoperimetric inequalities giving upper bounds on the area of discs with specified boundary and minimal area. The bounds are given as a function of the length of the boundary loop, and the function [0, ∞) → [0, ∞) giving the optimal bound is called F ill M 0 . Analogously, one can measure the complexity of the word problem in finitely presented groups by seeking isoperimetric inequalities giving upper bounds on the number of relators which one must apply in order to show that a word w in the given generators represents the trivial element in the group. The bounds are given in terms of the length of w, and the function N → N describing the optimal bound is called the Dehn function of the presentation.The Dehn functions associated to different presentations of a group are equivalent under the relation generated by composing functions with

Various notions of dimension for discrete groups are compared. A group is exhibited that acts with finite stabilizers on an acyclic 2-complex in such a way that the fixed point subcomplex for any non-trivial finite subgroup is contractible, but such that the group does not admit any such action on a contractible 2-complex. This group affords a counterexample to a natural generalization of the Eilenberg-Ganea conjecture.