Homological and homotopical higher-order filling functions

Young, Robert M.

doi:10.4171/ggd/144

Cited by 19 publications

(21 citation statements)

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“…If dim M D 2 then the same statement holds; this is proved below in Lemma 7.4. The case dim M D 3 is different: Young [23] has constructed a group G such that if M is a 3-manifold with boundary S 1 S 1 , then ı M .x/ is strictly larger than ı .2/ .x/.…”

Section: More General Dehn Functionsmentioning

confidence: 99%

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Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra

Brady

Bridson²,

Forester³

et al. 2009

Geom. Topol.

125

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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...

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Section: More General Dehn Functionsmentioning

confidence: 99%

“…This is not the case in higher dimensions, however. For each k > 2, Young [23] constructs a group for which ı .k/ .x/ is not subrecursive.…”

Section: Introductionmentioning

confidence: 99%

Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra

Brady

Bridson²,

Forester³

et al. 2009

Geom. Topol.

125

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“…Corollary 3, for a (simplicial) version of δ k . This is equivalent to the above (compare [48]). The second claim follows from [24], 2.A', for upper bounds and [10], 2.6.…”

Section: The Proof Of Theoremmentioning

confidence: 78%

“…We emphasize that Young constructed examples of groups Γ acting geometrically on a space X for which δ 2 Γ ∼ FV 3 X (see [48], Corollary 6). If, however, X = S × E × B is as in Theorem 1, then we also have δ 2 X ≻ FV 3 X .…”

Section: The Proof Of Theoremmentioning

confidence: 99%

Optimal higher-dimensional Dehn functions for some CAT(0) lattices

Leuzinger¹

2014

Groups Geom. Dyn.

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Let X = S × E × B be the metric product of a symmetric space S of noncompact type, a Euclidean space E and a product B of Euclidean buildings. Let Γ be a discrete group acting isometrically and cocompactly on X . We determine a family of quasi-isometry invariants for such Γ, namely the k-dimensional Dehn functions, which measure the difficulty to fill k-spheres by (k + 1)-balls (for 1 ≤ k ≤ dim X − 1). Since the group Γ is quasi-isometric to the associated CAT(0) space X , assertions about Dehn functions for Γ are equivalent to the corresponding results on filling functions for X . Basic examples of groups Γ as above are uniform S-arithmetic subgroups of reductive groups defined over global fields. We also discuss a (mostly) conjectural picture for non-uniform Sarithmetic groups.

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“…Remark 3.3. These notions are equivalent to the ones defined in [BBFS,You1] using admissible maps, as well as to the ones in [AWP], and those in [BH, p.153], [Bri2], [Ril,§2.3] that are using more polyhedra than just simplices.…”

Section: Higher Dimensional Isoperimetric Functionsmentioning

confidence: 99%

Combinatorial higher dimensional isoperimetry and divergence

Behrstock

Druţu²

2019

J. Topol. Anal.

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In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called "round" and "unfolded", provided that a bounded quasi-geodesic combing exists. We prove that the problem of estimating higher dimensional divergence as well can be restricted to round spheres. Applications of these results include a combinatorial analogy of the Federer-Fleming inequality for finitely generated groups, the construction of examples of CAT (0)-groups with higher dimensional divergence equivalent to x d for every degree d [BD2], and a proof of the fact that for bi-combable groups the filling function above the quasi-flat rank is asymptotically linear [BD3].

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Homological and homotopical higher-order filling functions

Abstract: Abstract. We construct groups in which FV 3 .n/ oe ı 2 .n/. This construction also leads to groups G k , k 3, for which ı k .n/ is not subrecursive.Mathematics Subject Classification (2010). 20F65, 57M07.

Cited by 19 publications

References 9 publications

Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra

Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra

Optimal higher-dimensional Dehn functions for some CAT(0) lattices

Combinatorial higher dimensional isoperimetry and divergence

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