${\rm CAT(0)}$ cube complexes are determined by their boundary cross ratio

Beyrer, Jonas; Fioravanti, Elia; Incerti-Medici, Merlin

doi:10.48550/arxiv.1805.08478

Cited by 2 publications

(3 citation statements)

References 22 publications

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“…which shows independence of the choice of the basepoint v in this case. In general, basepoint-independence follows from the following result, which we obtained in previous work with Incerti-Medici (Proposition 3.2 in [BFIM19]).…”

Section: The Cross Ratio On the Roller Boundarymentioning

confidence: 60%

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Cross ratios on ${\rm CAT(0)}$ cube complexes and marked length-spectrum rigidity

Beyrer,

Fioravanti

2019

Preprint

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We show that group actions on irreducible CAT(0) cube complexes with no free faces are uniquely determined by their ℓ 1 length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first length-spectrum rigidity result in a setting of non-positive curvature (with the exception of some particular cases in dimension 2 and symmetric spaces).As our main tool, we develop a notion of cross ratio on Roller boundaries of CAT(0) cube complexes. Inspired by results in negative curvature, we give a general framework reducing length-spectrum rigidity questions to the problem of extending cross-ratio preserving maps between (subsets of) Roller boundaries. The core of our work is then to show that, when there are no free faces, these cross-ratio preserving maps always extend to cubical isomorphisms. All our results equally apply to cube complexes with variable edge lengths.As a special case of our work, we construct a compactification of the Charney-Stambaugh-Vogtmann Outer Space for the group of untwisted outer automorphisms of an (irreducible) right-angled Artin group. This generalises the length function compactification of the classical Culler-Vogtmann Outer Space.1 See Theorem 5.1 in [Bis15]. 2 Besides the "obvious" cases where marked length-spectrum rigidity already follows from stronger forms of rigidity (e.g. symmetric spaces and Margulis' superrigidity).

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Section: The Cross Ratio On the Roller Boundarymentioning

confidence: 60%

“…We now go into more detail. Given vertices x, y, z, w ∈ X, we defined their cross ratio in previous work with Incerti-Medici [BFIM19]:…”

Section: Remarksmentioning

confidence: 99%

Cross ratios on ${\rm CAT(0)}$ cube complexes and marked length-spectrum rigidity

Beyrer,

Fioravanti

2019

Preprint

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“…In our setting, this space is always compact and totally disconnected -unlike the Gromov/visual 6 boundary ∂ ∞ X. As we observed in [BFIM19,BF19], the Roller boundary is naturally endowed with a continuous, Z-valued cross ratio:…”

Section: Introductionmentioning

confidence: 97%

Cross ratios and cubulations of hyperbolic groups

Beyrer,

Fioravanti

2018

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Many geometric structures associated to a Gromov hyperbolic group G can be encoded in terms of G-invariant cross ratios on the Gromov boundary ∂∞G. For instance, for surface groups these include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cubulations of arbitrary Gromov hyperbolic groups. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on ∂∞G.Our techniques also show that cubulations of hyperbolic groups are length-spectrum rigid (i.e. fully determined by their length function) if and only if they are essential and hyperplane-essential. This improves on our previous work in [BF19] and can be viewed as a proof of the marked length-spectrum rigidity conjecture in the cubical context.Along the way, we describe the relationship between the Roller boundary of a CAT(0) cube complex, its Gromov boundary and -in the nonhyperbolic case -the contracting boundary of Charney and Sultan. In fact, we show that even cubulations of non-hyperbolic groups G often give rise to canonical cross ratios on the contracting boundary ∂cG.All our results hold for cube complexes with variable edge lengths.

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${\rm CAT(0)}$ cube complexes are determined by their boundary cross ratio

Cited by 2 publications

References 22 publications

Cross ratios on ${\rm CAT(0)}$ cube complexes and marked length-spectrum rigidity

Cross ratios on ${\rm CAT(0)}$ cube complexes and marked length-spectrum rigidity

Cross ratios and cubulations of hyperbolic groups

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