Cross ratios and cubulations of hyperbolic groups

Abstract

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 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 ${\partial }_{\infty }G$. A consequence of our results is that essential, hyperplane-essential 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 in arXiv:1903.02447. Along the way, we describe the relationship between the Roller boundary of a $\mathrm{CAT}(0)$ 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.

47 pages, 4 figures. V4: minor modifications, mostly in the introductions. V3: improved introduction and abstract; some simplifications in Section 4. V2: This is in large part a new paper, now also including a marked length-spectrum rigidity result. All the material in Section 4 is new and this allows us to weaken hypotheses to the optimal ones

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• Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity
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Cited by in corpus (6)

• Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity
• $\mathrm{CAT}(0)$ cube complexes are determined by their boundary cross ratio
• Coarse-median preserving automorphisms
• Large facing tuples and a strengthened sector lemma
• Deforming cubulations of hyperbolic groups
• Coarse cubical rigidity