Top-dimensional quasiflats in CAT(0) cube complexes

Huang, Jingyin

doi:10.2140/gt.2017.21.2281

Cited by 26 publications

(22 citation statements)

References 32 publications

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“…Corollary F is related to [14,Lemma 4.9] and to statements in [5,25] about Euclidean sectors in cocompact CAT(0) cube complexes and arcs in the Tits boundary. In particular it shows that in any CAT(0) cube complex with a proper cocompact group action, non-trivial geodesic arcs on the Tits boundary can be extended to arcs of length at least π/2.…”

Section: Applications Of Theorem B To Boundaries Of Xmentioning

confidence: 84%

“…Many results follow from studying the geometry of CAT(0) cube complexes, often using strong properties reminiscent of negative curvature. For instance, several authors have studied the structure of quasi-flats and Euclidean sectors in cube complexes, with applications to rigidity properties of right-angled Artin group [5,14,25]. These spaces have also been shown to be median [8] and to have only semi-simple isometries [13].…”

Section: Introductionmentioning

confidence: 99%

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Retraction: Hierarchical hyperbolicity of all cubical groups

2017

Bulletin of London Math Soc

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Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. Then X has a factor system in the sense of Behrstock, Hagen and Sisto (Preprint, 2014). By results in Behrstock, Hagen and Sisto (Preprint, 2014), it follows that G is a hierarchically hyperbolic group; this answers questions raised in Behrstock, Sisto (Preprint, 2014, 2015). Our results also resolve a conjecture of Behrstock-Hagen on boundaries of cube complexes. In particular, X cannot contain a convex staircase.

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Section: Applications Of Theorem B To Boundaries Of Xmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Retraction: Hierarchical hyperbolicity of all cubical groups

2017

Bulletin of London Math Soc

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“…Furthermore, it follows easily that every n-dimensional quasiflat in a nonpositively curved symmetric space of rank n ≥ 2 lies within uniformly bounded distance from the union of a finite, uniformly bounded number of n-flats; compare Theorem 1.2.5 in [53] and Theorem 1.1 in [33]. We also refer to [10,13,47,49,62] for various similar statements.…”

Section: Proposition 101 (Lipschitz Extension)mentioning

confidence: 95%

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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“…Corollary E is related to Lemma 4.9 of [Hua14] and to statements in [Xie05,BKS08] about Euclidean sectors in cocompact CAT(0) cube complexes and arcs in the Tits boundary. In particular it shows that in any CAT(0) cube complex with a proper cocompact group action satisfying NICC, nontrivial geodesic arcs on the Tits boundary extend to arcs of length π/2.…”

Section: Introductionmentioning

confidence: 85%

“…Many results follow from studying the geometry of CAT(0) cube complexes, often using strong properties reminiscent of negative curvature. For instance, several authors have studied the structure of quasiflats and Euclidean sectors in cube complexes, with applications to rigidity properties of right-angled Artin group [Xie05,BKS08,Hua14]. These spaces have also been shown to be median [Che00] and to have only semi-simple isometries [Hag07].…”

Section: Introductionmentioning

confidence: 99%

On hierarchical hyperbolicity of cubical groups

Hagen

Susse

2020

Isr. J. Math.

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Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. We give three conditions on the action, any one of which ensures that X has a factor system in the sense of [BHS14]. We also prove that one of these conditions is necessary. This combines with [BHS14] to show that G is a hierarchically hyperbolic group; this partially answers questions raised in [BHS14,BHS15]. Under any of these conditions, our results also affirm a conjecture of Behrstock-Hagen on boundaries of cube complexes, which implies that X cannot contain a convex staircase. The necessary conditions on the action are all strictly weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that does not satisfy at least one of the conditions.

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Top-dimensional quasiflats in CAT(0) cube complexes

Cited by 26 publications

References 32 publications

Retraction: Hierarchical hyperbolicity of all cubical groups

Retraction: Hierarchical hyperbolicity of all cubical groups

Higher rank hyperbolicity

On hierarchical hyperbolicity of cubical groups

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