Prime orthogeodesics, concave cores and families of identities on hyperbolic surfaces.(English)Zbl 07966463
Summary: We prove and explore a family of identities relating lengths of curves and orthogeodesics of hyperbolic surfaces. These identities hold over a large space of metrics including ones with hyperbolic cone points, and in particular, show how to extend a result of the first author to surfaces with cusps. One of the main ingredients in the approach is a partition of the set of orthogeodesics into sets depending on their dynamical behavior, which can be understood geometrically by relating them to geodesics on orbifold surfaces. These orbifold surfaces turn out to be exactly on the boundary of the space in which the underlying identity holds.
MSC:
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 30F60 | Teichmüller theory for Riemann surfaces |
| 37E35 | Flows on surfaces |
| 53C22 | Geodesics in global differential geometry |
Keywords:
moduli spaces of hyperbolic surfaces;orthogeodesics;concave cores;immersed pairs of pants;trace equalitiesCite
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