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http://dx.doi.org/10.4134/BKMS.2015.52.4.1305

HYPERBOLIC NOTIONS ON A PLANAR GRAPH OF BOUNDED FACE DEGREE  

OH, BYUNG-GEUN (DEPARTMENT OF MATHEMATICS EDUCATION HANYANG UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.52, no.4, 2015 , pp. 1305-1319 More about this Journal
Abstract
We study the relations between strong isoperimetric inequalities and Gromov hyperbolicity on planar graphs, and give an alternative proof for the following statement: if a planar graph of bounded face degree satisfies a strong isoperimetric inequality, then it is Gromov hyperbolic. This theorem was formerly proved in the author's paper from 2014 [12] using combinatorial methods, while geometric approach is used in the present paper.
Keywords
planar graph; strong isoperimetric inequality; Gromov hyperbolicity;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 A. D. Aleksandrov and V. A. Zalgaller, Intrinsic Geometry of Surfaces, AMS Transl. Math. Monographs, v. 15, Providence, RI, 1967.
2 M. Bonk, Quasi-geodesic segments and Gromov hyperbolic spaces, Geom. Dedicata 62 (1996), no. 3, 281-298.   DOI
3 J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195-199. Princeton Univ. Press, Princeton, N. J., 1970.
4 M. Coornaert, T. Delzant, and A. Papadopoulos, Geometrie et theorie des groupes, LNM, Vol. 1441, Springer, Berlin, 1990.
5 E. Ghys and P. de la Harpe (eds.), Sur les Groupes Hyperbolique d'apres Mikhael Gromov, Birkhauser, Boston, MA, 1990.
6 M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), pp. 183-213, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981.
7 M. Gromov, Hyperbolic groups, In: Essays in group theory, 75-263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.
8 Z. He and O. Schramm, Hyperbolic and parabolic packings, Discrete Comput. Geom. 14 (1995), no. 2, 123-149.   DOI
9 M. Kanai, Rough isometries, and combinatorial approximations of geometries of non-compact Riemannian manifolds, J. Math. Soc. Japan 37 (1985) no. 3, 391-413.   DOI
10 B. Oh, Aleksandrov surfaces and hyperbolicity, Trans. Amer. Math. Soc. 357 (2005), no. 11, 4555-4577.   DOI   ScienceOn
11 B. Oh, Linear isoperimetric inequality and Gromov hyperbolicity on Aleksandrov surfaces, J. Chungcheong Math. Soc. 23 (2010), no. 2, 369-381.
12 B. Oh, Duality properties of strong isoperimetric inequalities on a planar graph and combinatorial curvatures, Discrete Comput. Geom. 51 (2014), no. 4, 859-884.   DOI   ScienceOn
13 Yu. G. Reshetnyak, Two-dimensional manifolds of bounded curvature, In: Geometry IV. Encyclopaedia of Mathematical Sciences (Yu. G. Reshetnyak eds.), pp. 3-163, Vol. 70, Springer, Berlin, 1993.
14 P. Soardi, Potential Theory on Infinite Networks, LNM 1590, Springer-Verlag, Berlin, 1994.