The asymptotic rank of metric spaces

Wenger, Stefan

doi:10.4171/cmh/223

Cited by 15 publications

(28 citation statements)

References 27 publications

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“…This is shown in a more general form, for complete metric spaces, and without restrictions on spt(Z), in Theorem 1.2 in [81]. The stated version suffices for our purposes, and the proof could be slightly simplified under these assumptions.…”

Section: Asymptotic Rankmentioning

confidence: 96%

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Higher rank hyperbolicity

Kleiner

Lang

2020

Invent. math.

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The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results for spaces of rank n ≥ 2 in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. For this purpose we replace quasi-geodesic lines with quasi-minimizing (locally finite) n-cycles of r n volume growth; prime examples include n-cycles associated with n-quasiflats. Solving an asymptotic Plateau problem and producing unique tangent cones at infinity for such cycles, we show in particular that every quasi-isometry between two proper CAT(0) spaces of asymptotic rank n extends to a class of (n − 1)-cycles in the Tits boundaries.(CI n ) (Coning inequalities) There is a constant c such that any two pointsx, x ′ in X can be joined by a curve of length ≤ c d(x, x ′ ), and for k = 1, . . . , n, every k-cycle R in some r-ball bounds a (k + 1)-chain S with massHere, for a general proper metric space X, we use metric integral currents (see Section 2). However, if X is bi-Lipschitz homeomorphic to a finite-dimensional simplicial complex with standard metrics on the simplices, then (by a variant of the Federer-Fleming deformation theorem [36]) one may equivalently take simplicial chains or singular Lipschitz chains (with integer coefficients). (AR n ) (Asymptotic rank ≤ n) No asymptotic cone of X contains an isometric copy of an (n + 1)-dimensional normed space. Equivalently, asrk(X) ≤ n, where asrk(X) is defined as the supremal k for which there exist a sequence r i → ∞ and subsets Y i ⊂ X such that the rescaled sets (Y i , r −1 i d) converge in the Gromov-Hausdorff topology to the unit ball in some k-dimensional normed space (see Section 4)., the required inequality holds for the geodesic cone S from the center of the r-ball over R (see Section 2.7). Furthermore, any n-connected simplicial complex as above with a properly discontinuous and cocompact simplicial action of a combable group satisfies (CI n ); see Section 10.2 in [32]. Every combable group, in particular every automatic group, admits such an action.When X is a cocompact CAT(0) or Busemann space, the asymptotic rank asrk(X) equals the maximal dimension of an isometrically embedded Euclidean or normed space, respectively [52]. More generally, for spaces satisfying (CI n ), condition (AR n ) is equivalent to a sub-Euclidean isoperimetric inequality for n-cycles [81]; this result, restated in Theorem 4.4, plays a key role in this paper. If X is a geodesic Gromov hyperbolic space, then every asymptotic cone of X is an R-tree, thus asrk(X) ≤ 1. Conversely, a space satisfying (CI 1 ) and (AR 1 ) is Gromov hyperbolic (compare Corollary 1.3 in [81] and the special case n = 1 of Theorem 1.1 below).We remark that the asymptotic rank is a quasi-isometry invariant for metric spaces [81], whereas condition (CI n ) is preserved, for instance, by quasiisome...

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Section: Asymptotic Rankmentioning

confidence: 96%

“…In this section we will first discuss the notion of asymptotic rank and the sub-Euclidean isoperimetric inequality from [81]. Then we will derive a localized version of this result as well as various characterizations of quasiminimizing local n-cycles in spaces of asymptotic rank at most n.…”

Section: Asymptotic Rankmentioning

confidence: 99%

Higher rank hyperbolicity

Kleiner

Lang

2020

Invent. math.

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“…In what follows, we work in somewhat greater generality than would be needed. This will allow us to prove new isoperimetric estimates in [19] which generalize those in [9] and [15].…”

Section: A Decomposition Theorem For Integral Currentsmentioning

confidence: 94%

“…In [14], Sormani and the author have recently exhibited sufficient conditions in terms of the topology of spt T n which imply equality in (2). A first application of our main result has recently been exhibited in [19], where Theorem 1.2 is used to prove sub-Euclidean isoperimetric inequalities for k-cycles in metric spaces of asymptotic rank k, including CAT(0)-spaces of Euclidean rank k.…”

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confidence: 86%

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Compactness for manifolds and integral currents with bounded diameter and volume

Wenger

2010

Calc. Var.

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By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class of oriented k-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio-Kirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which however we only assume uniform bounds on volume and diameter.

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Asymptotic rank of spaces with bicombings

Descombes

2016

Math. Z.

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The question, under what geometric assumptions on a space X an n-quasiflat in X implies the existence of an n-flat therein, has been investigated for a long time. It was settled in the affirmative for Busemann spaces by Kleiner, and for manifolds of non-positive curvature it dates back to Anderson and Schroeder. We generalize the theorem of Kleiner to spaces with bicombings. This structure is a weak notion of non-positive curvature, not requiring the space to be uniquely geodesic. Beside a metric differentiation argument, we employ an elegant barycenter construction due to Es-Sahib and Heinich by means of which we define a Riemannian integral serving us in a sort of convolution operation.

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The asymptotic rank of metric spaces

Cited by 15 publications

References 27 publications

Higher rank hyperbolicity

Higher rank hyperbolicity

Compactness for manifolds and integral currents with bounded diameter and volume

Asymptotic rank of spaces with bicombings

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