It is known that the universal Teichmuller space $T(1)=\{quasisymmetric \ homeomorphisms \ of \ S^1 \}/ SL (2, \mathbb R)$ is a group. My question is, under what conditions does the Teichmuller space $T(G)$ of a Fuchsian goup $G$ which is finitely generated and of the first kind a group. Or, basically, e.g., (under what conditions does it ture that:) if a quasiconformal homeomorphism $f: \mathbb H \to \mathbb H$ is compatible with $G$ then is $f^{-1}$ also compatible with $G.$
I have checked the mathoverflow, and found the following question which is related to the above question: Conjugate Groups of (quasi) Fuchsian GroupsConjugate Groups of (quasi) Fuchsian Groups