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• W2020198774 abstract "Let H be the upper half-plane equipped with the hyperbolic metric |dz|/Imz, and Γ a Fuchsian group acting on H. A subset S ⊂ H is said to be precisely invariant under a subgroup ΓS of Γ if γ(S) = S for all γ ∈ ΓS and γ(S)∩S = ∅ for all γ ∈ Γ−ΓS. Furthermore a hyperbolic element γ ∈ Γ is said to be simple if the axis of γ is precisely invariant under the cyclic subgroup 〈γ〉 generated by γ. The hyperbolic distance on H is denoted by d. For a quasiconformal automorphism f of H, the maximal dilatation of f is denoted by K(f). Let T (Γ) be the Teichmuller space of Γ and Mod(Γ) the Teichmuller modular group of Γ. It is known that if Γ is finitely generated of the first kind, then T (Γ) is finite dimensional and the action of Mod(Γ) on T (Γ) is properly discontinuous. This means that for every sequence {fn}n=1 of quasiconformal automorphisms of H satisfying fn ◦ Γ ◦ f−1 n = Γ and limn→∞K(fn) = 1, there exist an integer N and a conformal automorphism f of H such that fn are coincident with f on the real axis R for all n ≥ N . On the other hand, if Γ is infinitely generated, then T (Γ) is infinite dimensional and the action of Mod(Γ) is not properly discontinuous, in general. This means that there exists a sequence {fn}n=1 of quasiconformal automorphisms fn of H such that fn ◦ Γ ◦ f−1 n = Γ and limn→∞K(fn) = 1. On the basis of this fact, in [3], we proved that if Γ satisfies a certain bound condition on translation length, then Mod(Γ) acts properly discontinuously. The following proposition, which gives a lower bound of the maximal dilatation of a quasiconformal automorphism, is crucial for the proof." @default.
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• W2020198774 date "2005-06-01" @default.
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• W2020198774 title "A lower bound of maximal dilatations of quasiconformal automorphisms acting on Fuchsian groups" @default.
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• W2020198774 doi "https://doi.org/10.2996/kmj/1123767004" @default.
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