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Computing a Dirichlet domain for a hyperbolic surface

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arxiv 2212.01934 v1 pith:4YMOJXMY submitted 2022-12-04 cs.CG math.DGmath.GT

classification cs.CGmath.DGmath.GT
keywords surfacealgorithmdirichletdomaingeometryhyperbolicanalyzeclosed
verification ladder T0 review T1 audit T2 compute T3 formal
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The goal of this paper is to exhibit and analyze an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm finishes in polynomial time, in terms of the initial perimeter and the genus of the surface.

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Cited by 1 Pith paper

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  1. On dynamics of the Mapping class group action on relative $\text{PSL}(2,\mathbb{R})$-Character Varieties

    math.GT 2025-02 conditional novelty 6.0 of 10

    Simple-stable representations of punctured surface groups into PSL(2,R) form a domain of discontinuity for the mapping class group action, and admissible hyperbolic cone surface holonomies are simple-stable, with one-...

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