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The asymptotic geometry of the Teichmüller metric

Cormac Walsh 1, 2, 3 
1 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
3 TROPICAL - TROPICAL
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : We determine the asymptotic behaviour of extremal length along arbitrary Teichmüller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichmüller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the set of Busemann points. We show that the Busemann points are exactly the limits of the Teichmüller rays, and we give a necessary and sufficient condition for a sequence of Busemann points to converge to a Busemann point. Finally, we determine the detour metric on the boundary.
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https://hal.inria.fr/hal-00778085
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Submitted on : Friday, January 18, 2013 - 4:31:43 PM
Last modification on : Friday, July 8, 2022 - 10:04:20 AM

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Cormac Walsh. The asymptotic geometry of the Teichmüller metric. Geometriae Dedicata, Springer Verlag, 2019, 200 (1), pp.115-152. ⟨10.1007/s10711-018-0364-z⟩. ⟨hal-00778085⟩

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