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

Beyrer, Jonas; Fioravanti, Elia

doi:10.48550/arxiv.1903.02447

Cited by 6 publications

(31 citation statements)

References 35 publications

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“…Along with our previous work in [BF19], Theorem C also yields the following analogue of Theorem A in the context of non-hyperbolic groups acting on CAT(0) cube complexes with no free faces.…”

Section: Introductionmentioning

confidence: 55%

“…In this perspective, Theorem B is particularly interesting as -along with our previous work in [BF19] -it is the first length-spectrum rigidity result to cover such a broad family of non-positively curved spaces. We stress that the core arguments in the proof of Theorem B are completely different from those in [BF19]; see the discussion on Theorem E below.…”

Section: Introductionmentioning

confidence: 92%

“…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%

“…Note that, by Theorem A in [BF19], hyperbolicity, properness and cocompactness can be dropped from the statement of Theorem B, as long as we replace essentiality and hyperplane-essentiality with the stronger requirement that X and Y have no free faces. The advantage of the formulation of Theorem B presented above, however, is that it is the optimal result for cubulations of hyperbolic groups.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Cross ratios and cubulations of hyperbolic groups

Beyrer,

Fioravanti

2018

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“…(b) It is not known if (isometric) actions on finite-rank median spaces are completely determined by their length function. There are results of this type for actions on R-trees [CM87] and cube complexes [BF19b,BF19a], but their extension to a general median setting would require some significantly new ideas. The proof of Theorem E is made up of two main steps, which we now describe.…”

Section: Figurementioning

confidence: 99%

Coarse-median preserving automorphisms

Fioravanti¹

2021

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This paper has three main goals. First, we study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If ϕ is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that Fix ϕ is finitely generated and undistorted. Up to replacing ϕ with a proper power, we show that Fix ϕ acts properly and cocompactly on a convex subcomplex of the universal cover of the Salvetti/Davis complex. Thus, Fix ϕ is a special group in the sense of Haglund-Wise.By contrast, there exist "twisted" automorphisms of RAAGs for which Fix ϕ is undistorted but not of type F (hence not special), of type F but distorted, or even infinitely generated.Secondly, we introduce the notion of "coarse-median preserving" automorphism of a coarse median group, which plays a key role in the above results. We show that automorphisms of RAAGs are coarse-median preserving if and only if they are untwisted. On the other hand, all automorphisms of Gromov-hyperbolic groups and right-angled Coxeter groups are coarse-median preserving.Finally, we show that, for every special group G (in the sense of Haglund-Wise), every infiniteorder, coarse-median preserving outer automorphism of G can be realised as a homothety of a finite-rank median space X equipped with a "moderate" isometric G-action. This generalises the classical result, due to Paulin, that every infinite-order outer automorphism of a hyperbolic group H projectively stabilises a small H-tree.

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