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Stability of Gromov hyperbolicity. (English) Zbl 1205.30039

The authors explore the hyperbolicity, in the sense of Gromov, of flute surfaces, and consequently some trains. Trains and flute surfaces are special classes of Riemann surfaces, that are isomorphic to the complement in \(\mathbb{C}\) of a collection of intervals or points on the real line (with some supplementary conditions). They are endowed with their Poincaré metric. Recall it is a Riemannian metric of constant negative curvature obtained via identification of their universal cover with the Poincaré upper half plane. They are characterized by the sequence of lengths of a specific set of closed geodesics.
After giving definitions, basic properties, and recalling some of their previous results, the authors define the set \(H\) of sequences of lengths for which the corresponding flute surface is hyperbolic. They then give a list of 7 transformations on these sequences that leave \(H\) invariant, and prove they do. The proof is based on a characterization of flute surfaces that is done in another article submitted by the authors.

MSC:

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
30C99 Geometric function theory
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