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Article Dans Une Revue Expositiones Mathematicae Année : 2021

From Euclidean triangles to the hyperbolic plane

Ismail Saglam
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Résumé

We introduce a model of the hyperbolic plane which arises from Thurston's ideas on best Lipschitz maps between surfaces. More precisely, we apply Thurston's theory to the space of Euclidean triangles. We prove that the Lipschitz constant of the unique affine map between two Euclidean triangles induces a metric on the set of isometry classes of the triangles with area 1/ 2. We then prove that this space of triangles together with 2 times its natural metric is isometric to the hyperbolic plane. Our construction is simple and natural and it makes relations between several geometrical ideas. This paper will appear in the journal Expositiones Mathematicae.
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Dates et versions

hal-03419775 , version 1 (08-11-2021)

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Ismail Saglam. From Euclidean triangles to the hyperbolic plane. Expositiones Mathematicae, In press, ⟨10.1016/j.exmath.2021.10.003⟩. ⟨hal-03419775⟩
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