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We compare Green’s function g on an infinite volume, hyperbolic Riemann surface X with an analogous discrete function gdisc on a graphical caricature Γ of X. The main result, modulo technical hypotheses, is that g and gdisc differ by at most an additive constant C which depends only on the Euler characteristic of X. In particular, the estimate of g by gdisc remains uniform as the geometry (i.e. the conformal structure) of X varies.

Green's functions, electric networks, and the geometry of hyperbolic Riemann surfaces