My question are related with this paragraph: (page 25 of A Fibre Bundle Description of TeichÃ¼ller Theory by Earle and Eells.)

Let $X$ a Riemann surface of genus g>1, if $J\in M(X)$ ($M(X)$ space of complex structures in $X$), then we have $\pi:U\rightarrow X$ smooth covering map ($U$ the upper half plane) whose cover group $\Gamma$ is Fuchsian. Then $\pi$ induces $\pi^{*}:M(X)\rightarrow M(U)$ whose imagen is denoted by $M(\Gamma)$, its elements are the $\Gamma-$invariant complex structures.

1.- Why the elements of $M(\Gamma)$ are $\Gamma-$invariant complex structures.