On the Filling Invariants at Infinity of Hadamard Manifolds

Hindawi, Mohamad A.

doi:10.1007/s10711-005-9002-7

Cited by 4 publications

(4 citation statements)

References 12 publications

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“…Related results have been obtained for symmetric spaces of non-compact type ( [5], [23], [15]) and for proper cocompact Hadamard spaces ( [28]) for the higher divergence invariants div k of Brady and Farb. As mentioned above, symmetric spaces X of non-compact type admit linear isoperimetric inequalities for I k (X) for all k ≥ asrk(X).…”

Section: Introductionmentioning

confidence: 72%

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The asymptotic rank of metric spaces

Wenger¹

2011

Comment. Math. Helv.

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Abstract. In this article we define and study a notion of asymptotic rank for metric spaces and show in our main theorem that for a large class of spaces, the asymptotic rank is characterized by the growth of the higher isoperimetric filling functions. For a proper, cocompact, simply connected geodesic metric space of non-positive curvature in the sense of Alexandrov the asymptotic rank equals its Euclidean rank.Mathematics Subject Classification (2010). 49Q15.

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Section: Introductionmentioning

confidence: 72%

“…As a consequence of Theorems Related results have been obtained for symmetric spaces of non-compact type ( [6], [24], [16]) and for proper cocompact Hadamard spaces ( [29]) for the higher divergence invariants div k of Brady and Farb.…”

Section: Introductionmentioning

confidence: 86%

The asymptotic rank of metric spaces

Wenger¹

2011

Comment. Math. Helv.

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“…For k ≥ 0, our functions Div k are closely related to the higher divergence functions defined by Brady and Farb in [5] for the special case of Hadamard manifolds. Using the manifold definition, combined results of Leuzinger and Hindawi prove that the higher divergence functions detect the real-rank of a symmetric space, as Brady-Farb had conjectured [26,23]. Thus the geometry and the algebra are connected.…”

Section: Introductionmentioning

confidence: 93%

Pushing fillings in right-angled Artin groups

Abrams

Brady

Dani

et al. 2012

Journal of the London Mathematical Society

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Abstract. We define a family of quasi-isometry invariants of groups called higher divergence functions, which measure isoperimetric properties "at infinity." We give sharp upper and lower bounds on the divergence functions for right-angled Artin groups, using different pushing maps on the associated cube complexes. In the process, we define a class of RAAGs we call orthoplex groups, which have the property that their Bestvina-Brady subgroups have hard-to-fill spheres. Our results give sharp bounds on the higher Dehn functions of Bestvina-Brady groups, a complete characterization of the divergence of geodesics in RAAGs, and an upper bound for filling loops at infinity in the mapping class group.

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“…In dealing with isoperimetric functions some authors use an alternate formulation where the volume is bounded above by Ar, instead of Ar k ; this yields an equivalent notion (although the functions differ by a power of k), but with the alternative definition one must modify the equivalence relation to allow an additive term which is a multiple of r instead of r k . We use the present definition because it yields a formulation consistent with the standard definition of the divergence function that we use in this paper, see also [BF,Hin,Leu1,Wen2].…”

Section: Fillvol(h) Vol(h) Diam(h)mentioning

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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On the Filling Invariants at Infinity of Hadamard Manifolds

Cited by 4 publications

References 12 publications

The asymptotic rank of metric spaces

The asymptotic rank of metric spaces

Pushing fillings in right-angled Artin groups

Combinatorial higher dimensional isoperimetry and divergence

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