Cubulations and cross ratios on contracting boundaries


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_∞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 \mathrmCAT(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.


Research Groups: Groups and Geometry

Publication type: Journal

Journal: arXiV preprints

Citation: arXiv:1810.08087 [math.GT]

Date Published: 2018


Registered Mode: imported from a bibtex file

Authors: Jonas Beyrer, Elia Fioravanti

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Created: 30th Jan 2020 at 10:22

Last updated: 5th Mar 2024 at 21:24

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