On hierarchical hyperbolicity of cubical groups

Hagen, Mark F.; Susse, Tim

doi:10.1007/s11856-020-1967-2

Cited by 28 publications

(19 citation statements)

References 34 publications

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“…These include, among others, relative hyperbolicity [16,30,34,39,70], various notions of "directional" hyperbolicity inherent in stability/contraction properties of (quasi-)geodesics [12,21,23,48,51,75,76] (this in fact goes back to the notion of rank one geodesics [5,6] which predates hyperbolicity), acylindrical hyperbolicity [11,15,25,71], and hierarchical hyperbolicity [9,10,43,66]. (The literature is far richer than indicated here -we apologize for omissions.)…”

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confidence: 96%

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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confidence: 96%

Higher rank hyperbolicity

Kleiner

Lang

2020

Invent. math.

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“…Proof Let S denote the intersection of B = B(u, w) with the carrier of u and consider its stabiliser G S ≤ G. The action G S S is cocompact by Proposition 2.7 in [41]. There exists a constant D > 0 such that S is contained in the D-neighbourhood of both u and w. The finitely many hyperplanes that contain S in their D-neighbourhood are permuted by G S and it follows that G u ∩ G w sits in G S as a finite-index subgroup.…”

Section: Throughout This Subsectionmentioning

confidence: 99%

Cross ratios and cubulations of hyperbolic groups

Beyrer

Fioravanti

2021

Math. Ann.

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Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cocompact cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary $$\partial _{\infty }G$$ ∂ ∞ G . A consequence of our results is that essential, hyperplane-essential, cocompact cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work [4]. Along the way, we describe the relationship between the Roller boundary of a $$\mathrm{CAT(0)}$$ CAT ( 0 ) cube complex, its Gromov boundary and—in the non-hyperbolic case—the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

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“…Given a crooked hyperplane Γ ⊆ G(S ), we denote by G Γ the G-stabiliser of the subset † U (Γ) ⊆ X. Part ( 5) of [30,Proposition 3.8 and Lemma 2.3] shows that the action G Γ U (Γ) is proper and cocompact. By part (3), the subgroup G Γ G is quasiconvex.…”

Section: Systems Of Switchesmentioning

confidence: 99%

Deforming cubulations of hyperbolic groups

Fioravanti

Hagen

2021

Journal of Topology

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We describe a procedure to deform cubulations of hyperbolic groups by 'bending hyperplanes'. Our construction is inspired by related constructions like Thurston's Mickey Mouse example, walls in fibred hyperbolic 3-manifolds and free-by-Z groups, and Hsu-Wise turns. As an application, we show that every cocompactly cubulated Gromov-hyperbolic group admits a proper, cocompact, essential action on a CAT(0) cube complex with a single orbit of hyperplanes. This answers (in the negative) a question of Wise, who proved the result in the case of free groups. We also study those cubulations of a general group G that are not susceptible to trivial deformations. We name these bald cubulations and observe that every cocompactly cubulated group admits at least one bald cubulation. We then apply the hyperplane-bending construction to prove that every cocompactly cubulated hyperbolic group G admits infinitely many bald cubulations, provided G is not a virtually free group with Out(G) finite. By contrast, we show that the Burger-Mozes examples each admit a unique bald cubulation. Contents

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On hierarchical hyperbolicity of cubical groups

Cited by 28 publications

References 34 publications

Higher rank hyperbolicity

Higher rank hyperbolicity

Cross ratios and cubulations of hyperbolic groups

Deforming cubulations of hyperbolic groups

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