Abstract
A domain in the plane obtained by removing all integer lattice points admits the hyperbolic metric, which is the rank 2 Abelian cover of the once-punctured square tours. We compare the hyperbolic metric of this domain with a scaled Euclidean metric in the complement of the cusp neighborhoods. They are quasi-isometric. We investigate the best possible quasi-isometry constant relying on numerical experiment by Mathematica.
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References
F.F. Bonsall, J. Duncan, Numerical Ranges. II. London Mathematical Society Lecture Notes Series, vol. 10 (Cambridge University Press, Cambridge, 1973)
D. Mumford, C. Series, D. Wright, Indra’s Pearls. The Vision of Felix Klein (Cambridge University Press, Cambridge, 2002)
Acknowledgement
This work was supported by JSPS KAKENHI 25287021.
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Matsuzaki, K. (2017). The Hyperbolic Metric on the Complement of the Integer Lattice Points in the Plane. In: Dang, P., Ku, M., Qian, T., Rodino, L. (eds) New Trends in Analysis and Interdisciplinary Applications. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-48812-7_31
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DOI: https://doi.org/10.1007/978-3-319-48812-7_31
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-48810-3
Online ISBN: 978-3-319-48812-7
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