Deforming hyperbolic hexagons with applications to the arc and the Thurston metrics on Teichmüller spaces

Athanase Papadopoulos (1) , Sumio Yamada (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Department of Mathematics [Gakushuin]

For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy class of this map relative to the boundary. As a consequence of this construction, we exhibit new geodesics for the arc metric on the Teichmüller space of an arbitrary surface of negative Euler characteristic with nonempty boundary. We also obtain new geodesics for Thurston’s metric on Teichmüller spaces of hyperbolic surfaces without boundary. Our results generalize results obtained in the two papers by Papadopoulos and Théret (Proc. Am. Math. Soc. 138(5):1775--1784, 2010 and Geom. Dedic. 161:63--83, 2012).

Mots clés: Hyperbolic geometry; Teichmüller space; Arc metric; Geodesic; Lipschitz map; Thurston’s metric; Deforming hyperbolic hexagons

Domaines: Topologie géométrique [math.GT]; Géométrie métrique [math.MG]; Variables complexes [math.CV]

Athanase Papadopoulos, Sumio Yamada. Deforming hyperbolic hexagons with applications to the arc and the Thurston metrics on Teichmüller spaces. Monatshefte für Mathematik, 2017, April 2017, Volume 182, Issue 4, pp. 913--939, ⟨10.1007/s00605-017-1023-4⟩. ⟨hal-01516356⟩