Abstract
The fat flow modulus is a discrete version of the classical conformal modulus; one can use it to classify triangulations of half-open annuli as parabolic or hyperbolic. There exist various criteria for parabolicity; several of these criteria involve the vertex valences of the triangulation. In this paper we decompose the half-open annulus into a family of concentric sub-annuli. We can estimate the fat flow moduli of these sub-annuli in terms of their vertex valences. By using the Layer Theorem of Cannon et al. [2], we sum the estimates for these sub-annuli to prove a conjecture of He and Schramm [6]. The result is a new parabolicity criterion.
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Received April 14, 1999, and in revised form June 21, 2000. Online publication August 28, 2001.
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Repp, A. Bounded Valence Excess and the Parabolicity of Tilings. Discrete Comput Geom 26, 321–351 (2001). https://doi.org/10.1007/s00454-001-0040-0
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DOI: https://doi.org/10.1007/s00454-001-0040-0