Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra

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...

Non-positive curvature and complexity for finitely presented groups

“…We prove that if P is an irreducible non-negative integer matrix with Perron-Frobenius eigenvalue λ > 1, and r is an integer greater than every row sum in P , then for every k ≥ 2 there is a group Γ = Σ k−1 G r,P with a compact (k + 1)-dimensional classifying space such that δ (k) (x) x 2 log λ (r) . It follows from this and a related result in [16] that IP…”

“…These complexes are obtained by attaching a pair of annuli to a torus in a manner that ensures the existence of a family of discs in the universal cover that display a certain snowflake geometry. With Max Forester and Ravi Shankar [16], we developed a more sophisticated version of the snowflake construction that yields a much larger class of isoperimetric exponents, showing in particular that [2, ∞) ∩ Q ⊆ IP.…”

“…Until recently, our knowledge even for IP (2) was remarkably sparse, but my recent work with Brady, Forester and Shankar [16] remedies this. We prove that if P is an irreducible non-negative integer matrix with Perron-Frobenius eigenvalue λ > 1, and r is an integer greater than every row sum in P , then for every k ≥ 2 there is a group Γ = Σ k−1 G r,P with a compact (k + 1)-dimensional classifying space such that δ (k) (x) x 2 log λ (r) .…”

Homological and homotopical higher-order filling functions

“…In some cases, these functions are equivalent; for example, the methods used in [11] work for all these definitions. Along these lines, Brady et al [3] showed that if @M is connected and dim M D k C1 4 then ı M .n/ ı k .n/. In this note, we will show that this is not necessarily true if dim M D 3, and that there are groups where FV 3 is not equivalent to ı 2 .…”

“…To define ı k , we will take the approach of Brady et al [3], which is equivalent to the definition of Alonso, Wang, and Pride [2] or of Bridson [5]. We recall their definition of an admissible map: Definition 1 (Admissible maps [3]). Let W be a compact k-manifold and X a CWcomplex.…”