Erdős distinct distances in hyperbolic surfaces

Lu, Zhipeng; Meng, Xianchang

doi:10.48550/arxiv.2006.16565

Cited by 3 publications

(13 citation statements)

References 28 publications

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“…Similar to the argument for Theorem 1.1 of [5], this in turn establishes Guth-Katz type lower bound of Erdős distinct distances problem in hyperbolic surfaces corresponding to geometrically finite Fuchsian groups. To the complementary, in section 2.2 we show that Theorem 1.3.…”

Section: Introductionsupporting

confidence: 61%

“…This is the combination of Theorem 3.11 and Theorem 3.13, which addresses Conjecture 3 of [5]. Our proof is based on Shimizu's lemma introduced by Shimizu [6].…”

Section: Introductionmentioning

confidence: 83%

“…It will be interesting to see more precise computational results on geodesic covering numbers of geometrically finite groups and their applications to various geometric and combinatorial problems. The conjectured relation between those numbers and signatures of Fuchsian groups or Fenchel-Nielsen coordinates in the Techmüller space in [5] still calls for more studies, to which the current paper may have implications.…”

Section: Introductionmentioning

confidence: 87%

“…In the work of Lu-Meng [5], a notion called geodesic cover of Fuchsian groups is introduced to deal with the Erdős distinct distances problem in hyperbolic surfaces. There we initiate the investigation of quantitative aspects of the notion for special Fuchsian groups including the modular group and standard regular surface groups.…”

Section: Introductionmentioning

confidence: 99%

“…In [5], the geodesic covering number of the modular group PSL 2 (Z) is precisely estimated to be 4 in the first definition and 10 in the second. Also, those of standard regular hyperbolic surfaces of genus g (≥ 2) are estimated to establish lower bounds of Erdős distinct distances problem in such surfaces, see Theorem 1.2 and Proposition 3.1 of [5].…”

Section: Introductionmentioning

confidence: 99%

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Geodesic cover of Fuchsian groups

2021

Preprint

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We study unions of fundamental domains of a Fuchsian group, especially those with hyperbolic plane metric realizing the metric of the corresponding hyperbolic surface. We call these unions the geodesic covers of the Fuchsian group or the hyperbolic surface. The paper contributes to showing that finiteness of geodesic covers is basically another characterization of geometrically finiteness. The resolution of geometrically finite case is based on Shimizu's lemma.

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Section: Introductionsupporting

confidence: 61%

“…This is the combination of Theorem 3.11 and Theorem 3.13, which addresses Conjecture 3 of [5]. Our proof is based on Shimizu's lemma introduced by Shimizu [6].…”

Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Geodesic cover of Fuchsian groups

2021

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Computing distances on Riemann surfaces

Stepanyants,

Beardon,

Paton

et al. 2024

J. Phys. A: Math. Theor.

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Riemann surfaces are among the simplest and most basic geometric objects. They appear as key players in many branches of physics, mathematics, and other sciences. Despite their widespread significance, how to compute distances between pairs of points on compact Riemann surfaces is surprisingly unknown, unless the surface is a sphere or a torus. This is because on higher-genus surfaces, the distance formula involves an infimum over infinitely many terms, so it cannot be evaluated in practice. Here we derive a computable distance formula for a broad class of Riemann surfaces. The formula reduces the infimum to a minimum over an explicit set consisting of finitely many terms. We also develop a distance computation algorithm, which cannot be expressed as a formula, but which is more computationally efficient on surfaces with high genuses. We illustrate both the formula and the algorithm in application to generalized Bolza surfaces, which are a particular class of highly symmetric compact Riemann surfaces of any genus greater than 1.

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Distinct distances on hyperbolic surfaces

Meng¹

2020

Preprint

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For any cofinite Fuchsian group Γ ⊂ PSL(2, R), we show that any set of N points on the hyperbolic surface Γ\H 2 determines ≥ C Γ N log N distinct distances for some constant C Γ > 0 depending only on Γ. In particular, for Γ being any finite index subgroup of PSL(2, Z) with µ = [PSL(2, Z) : Γ] < ∞, any set of N points on Γ\H 2 determines ≥ C N µ log N distinct distances for some absolute constant C > 0.

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Erdős distinct distances in hyperbolic surfaces

Cited by 3 publications

References 28 publications

Geodesic cover of Fuchsian groups

Geodesic cover of Fuchsian groups

Computing distances on Riemann surfaces

Distinct distances on hyperbolic surfaces

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