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Rough isometries and Dirichlet finite harmonic functions on graphs

Author: Paolo M. Soardi
Journal: Proc. Amer. Math. Soc. 119 (1993), 1239-1248
MSC: Primary 31C05; Secondary 94C05
MathSciNet review: 1158010
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Abstract: Suppose that $ {G_1}$ and $ {G_2}$ are roughly isometric connected graphs of bounded degree. If $ {G_1}$ has no nonconstant Dirichlet finite harmonic functions, then neither has $ {G_2}$.

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  • [CW] D. I. Cartwright and W. Woess, Infinite graphs with nonconstant Dirichlet finite harmonic functions, SIAM J. Discrete Math. 3 (1992), 380-385. MR 1172746 (94a:31005)
  • [DS] P. G. Doyle and J. L. Snell, Random walks and electrical networks, Math. Assoc. Amer., Washington, D.C., 1984. MR 920811 (89a:94023)
  • [Du] R. J. Duffin, The extremal length of a network, J. Math. Anal. Appl. 5 (1962), 200-215. MR 0143468 (26:1024)
  • [G] P. Gerl, Random walks on graphs, Probability Measures on Groups VII (H. Heyer, ed.), Lecture Notes in Math., vol. 1210, Springer, New York, 1985, pp. 285-303. MR 879011 (88m:60178)
  • [Gr] M. Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), Springer, New York, 1987, pp. 75-263. MR 919829 (89e:20070)
  • [K] V. A. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Potential Anal. 1 (1992), 61-82. MR 1245225 (94i:31012)
  • [Ka1] M. Kanai, Rough isometries and combinatorial approximation of geometries of non-compact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), 391-413. MR 792983 (87d:53082)
  • [Ka2] -, Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan 38 (1986), 227-238. MR 833199 (87e:53066)
  • [KY] T. Kayano and M. Yamasaki, Boundary limits of discrete Dirichlet potentials, Hiroshima J. Math. 14 (1984), 401-406. MR 764458 (86j:31007)
  • [MMT] S. Markvorsen, S. Mc Guinness, and C. Thomassen, Transient random walks on graphs and metric spaces with applications to hyperbolic surfaces, Proc. London Math. Soc. 64 (1992), 1-20. MR 1132852 (93e:60142)
  • [S] P. M. Soardi, Recurrence and transience of the edge graph of a tiling of the euclidean plane, Math. Ann. 287 (1990), 613-626. MR 1066818 (92b:52044)
  • [SW] P. M. Soardi and W. Woess, Uniqueness of currents in infinite resistive networks, Discrete Appl. Math. 31 (1991), 37-49. MR 1097526 (92b:94052)
  • [SY] P. M. Soardi and M. Yamasaki, Classification of infinite networks and its applications, Circuits Systems Signal Process. 12 (1993), 133-149.
  • [T1] C. Thomassen, Resistances and currents in infinite electrical networks, J. Combin. Theory Ser. B 49 (1990), 87-102. MR 1056821 (91d:94029)
  • [T2] -, Isoperimetric inequalities and transient random walks on graphs (to appear).
  • [Y1] M. Yamasaki, Parabolic and hyperbolic infinite networks, Hiroshima Math. J. 7 (1977), 135-146. MR 0429377 (55:2395)
  • [Y2] -, Discrete potentials on an infinite network, Mem. Fac. Sci. Shimane Univ. 13 (1979), 31-44. MR 558311 (81h:31016)
  • [Y3] -, Ideal boundary limit of discrete Dirichlet functions, Hiroshima Math. J. 16 (1986), 353-360. MR 855163 (87m:31010)
  • [Z] A. H. Zemanian, Infinite electrical networks, Proc. IEEE 64 (1976), 6-17. MR 0453371 (56:11635)

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Keywords: Rough isometries, infinite networks, infinite graphs, Dirichlet finite harmonic functions
Article copyright: © Copyright 1993 American Mathematical Society