Cubulable Kähler groups

Delzant, Thomas; Py, Pierre

doi:10.2140/gt.2019.23.2125

Cited by 15 publications

(13 citation statements)

References 52 publications

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“…So it would also be interesting to determine under what circumstances an invariant Rvalued cross ratio can be discretised to an invariant Z-valued cross ratio. Of course, one should be very careful when making speculations, as, for instance, uniform lattices in SU (n, 1) and Sp(n, 1) are not cubulable [27,53].…”

Section: Theorem C Let a Non-elementary Gromov Hyperbolic Group G Act Properly And Cocompactly On Essential Hyperplane-essentialmentioning

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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Section: Theorem C Let a Non-elementary Gromov Hyperbolic Group G Act Properly And Cocompactly On Essential Hyperplane-essentialmentioning

confidence: 99%

Cross ratios and cubulations of hyperbolic groups

Beyrer

Fioravanti

2021

Math. Ann.

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“…The following result allows us to construct free subgroups of Γ; in the case of CAT(0) cube complexes, compare with Theorem 6 in [DP16] and Theorem F in [CS11]. Note that only part (1) is needed to prove the Tits alternative; we will use part (2) in [Fio17b] to characterise Roller elementarity in terms of the vanishing of a certain cohomology class.…”

Section: Caprace-sageev Machinerymentioning

confidence: 99%

The Tits alternative for finite rank median spaces

Fioravanti

2019

Enseign. Math.

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“…In the case of simplicial trees, it appears in [FV17]. For CAT(0) cube complexes, the implication "Roller nonelementary ⇒ [b] = 0" is implicit in [DP16]. In [CFI16], the authors construct a family of bounded cohomology classes detecting Roller elementarity in CAT(0) cube complexes.…”

Section: Introductionmentioning

confidence: 99%

Superrigidity of actions on finite rank median spaces

Fioravanti¹

2019

Advances in Mathematics

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Finite rank median spaces are a simultaneous generalisation of finite dimensional CAT(0) cube complexes and real trees. If Γ is an irreducible lattice in a product of rank-one simple Lie groups, we show that every action of Γ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces, where even proper cocompact actions can arise.The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated topological groups.We exploit Roller compactifications of median spaces; these were introduced in [Fio17] and generalise a well-known construction in the case of cube complexes. We show that the Haagerup cocycle provides a reduced 1-cohomology class that detects group actions with a finite orbit in the Roller compactification. This is new even for CAT(0) cube complexes and has interesting consequences involving Shalom's property HF D . For instance, in Gromov's density model, random groups at low density do not have HF D .

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Cubulable Kähler groups

Cited by 15 publications

References 52 publications

Cross ratios and cubulations of hyperbolic groups

Cross ratios and cubulations of hyperbolic groups

The Tits alternative for finite rank median spaces

Superrigidity of actions on finite rank median spaces

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