Strongly quasisymmetirc homeomorphisms being compatible with Fuchsian groups

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.