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...he assertion in line 2.4.1.1 of [4] which states that every cut contains a subcut which is a skinny cut. Let � R ′ denote the universal cover of R ′ with covering projection π : � R ′ → R ′ . Let α : =-=[0, 1]-=- → R be a τi-minimal simple closed curve in R separating the ends of R. Let �α : R → � R ′ denote a path for which π(�α(t + z)) = α(t) for every t ∈ [0, 1] and z ∈ Z. The shingles of S all lift to � R...

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...§0. Introduction. This paper involves the burgeoning field of discrete approximations to complex analysis and conformal mapping. The purpose is to improve the combinatorial Riemann mapping theorem of =-=[3]-=- by simplifying its hypotheses and thereby greatly enhancing its applicability. We recall here the purpose of the combinatorial Riemann mapping theorem by comparing it with the other discrete Riemann ...

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...fficiently large positive integer i. In this paragraph we determine a lower bound for the τi-height H(R, τi) of R. Let α be a τi-minimal path for R joining the ends of R. According to line 2.4.1.1 in =-=[4]-=-, the fat flow with underlying topological path α has a skinny subflow. Hence there exist shingles Dj1 , . . . , Djh in S all of which meet α such that Dj1 ∩ R0 �= ∅, Djl ∩ Djl+1 �= ∅ for l ∈ {1, . . ...

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...e joined to O by a geodesic segment which contains x. Two vertices x, y ∈ Γ are said to have the same cone type if left multiplication yx −1 : Γ → Γ takes C(x, O) isomorphically to C(y, O). By [1] or =-=[2]-=-, every negatively curved group has only finitely many cone types. Theorem 8.2. Let G be a negatively curved group. Let Γ be a locally finite Cayley graph for G with space at infinity ∂Γ = S 2 . Given...

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...LY RICH FAMILIES OF PLANAR RINGS 31 §8. A conformality criterion for negatively curved groups. In this section we apply the sufficiently rich theorem to a sequence of disks at infinity constructed in =-=[6]-=- coming from a negatively curved group whose space at infinity is the 2-sphere. We show in this situation that Axiom I and Axiom II can be replaced by Axiom 0. In fact the argument shows that it suffi...

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...original; (2) we replace the verification of the two conditions for all of an uncountable set of domains to the verification of the simpler axioms for a finite number of domains; (3) in another paper =-=[5]-=- we show that the simpler axiom can actually be verified in a variety of nontrivial situations provided there is substantial symmetry present (and symmetries, though unfortunately not of the kind we u...