Dirichlet domains and the NEC uniformization of extremal hyperbolic surfaces

Ernesto Girondo; Claudia Muñoz

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arXiv · 2607.15882

Dirichlet domains and the NEC uniformization of extremal hyperbolic surfaces

Ernesto Girondo·Claudia Muñoz

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Abstract

Extremal hyperbolic surfaces are finite-type hyperbolic surfaces that contain an embedded disc of the largest possible radius. We introduce a new general approach to these objects, based on the uniformization by NEC groups, obtaining a structural description of the Dirichlet domains associated to extremal disc centers for surfaces with cusps and/or geodesic boundary. We also provide a number of applications of our new description of extremal surfaces, as an effective counting formula for extremal disc configurations, the determination of every such configuration with Euler characteristic -1 and -2, the location of hidden extremal discs in several families of extremal surfaces, and the full description of their automorphism groups.

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Ernesto Girondo, Claudia Muñoz. 2026-07-17. Dirichlet domains and the NEC uniformization of extremal hyperbolic surfaces. https://arxiv.org/abs/2607.15882

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