Strongly quasisymmetirc homeomorphisms being compatible with Fuchsian groups

TL;DR

This paper characterizes strongly quasisymmetric homeomorphisms compatible with convergence Fuchsian groups of the first kind through the existence of quasiconformal extensions with measures satisfying Carleson measure conditions, extending to Carleson-Denjoy domains.

Contribution

It introduces a generalized Dirichlet fundamental domain for Fuchsian groups with parabolic elements and characterizes strongly quasisymmetric homeomorphisms via quasiconformal extensions with Carleson measure conditions.

Findings

Characterization of strongly quasisymmetric homeomorphisms via quasiconformal extensions.

Introduction of generalized Dirichlet fundamental domain for Fuchsian groups.

Extension of results to Carleson-Denjoy domains.

Abstract

In this paper we first introduced a domain called generalized Dirichlet fundamental domain $\mathcal{F}^{*}$ 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 $\mathbb{H}$ onto itself such that the induced measure $\lambda_{\mu}=|\mu|^{2}/Im(z)dxdy$ by the Beltrami coefficient $\mu$ of $f$ is a Carleson measure on the generalized Dirichlet fundamental domain $\mathcal{F}^{*}.$ We also show that the above property also holds for Carleson-Denjoy domains.