Strongly quasisymmetirc homeomorphisms being compatible with Fuchsian groups

2022/06/08 by Huo, Shengjin, Zhao, Mengzhen
#30F35 #30F60 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.03784

Abstract

In this paper we first introduced a domain called generalized Dirichlet fundamental domain F^* for a Fuchsian group G whose generators contain parabolic elements. This allows us to show that a quasisymmetric homeomorphism h being compatible with a convergence Fuchsian group G of first kind is a strongly quasisymmetric homeomorphism if and only if it has a quasiconformal extension f to the upper half plane ℍ onto itself such that the induced measure λ_μ=|μ|²/Im(z)dxdy by the Beltrami coefficient μ of f is a Carleson measure on the generalized Dirichlet fundamental domain F^*. We also show that the above property also holds for Carleson-Denjoy domains.

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