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- W4205744764 abstract "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 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 $$partial _{infty }G$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> . 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 $$mathrm{CAT(0)}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>CAT</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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." @default.
- W4205744764 created "2022-01-26" @default.
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- W4205744764 date "2021-12-11" @default.
- W4205744764 modified "2023-10-14" @default.
- W4205744764 title "Cross ratios and cubulations of hyperbolic groups" @default.
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- W4205744764 doi "https://doi.org/10.1007/s00208-021-02330-3" @default.
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