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My question are related with this paragraph: (page 25 of A Fibre Bundle Description of Teichüller 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.

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Your question 1 makes sense, but is perhaps suited to math.stackexchange.com, which has a broader remit than MO. Your question 2 is completely ungrammatical, and I can't figure out what you mean. Also, what does it mean for a space of complex structures to be unitary? –  David Roberts Sep 14 '11 at 2:19
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@daniel, the complex structures in the image of $\pi^*$ are pullbacks of complex structures on $X$ via $\pi$. Being the pullbacks, they are $\Gamma$-invariant. –  Igor Belegradek Sep 15 '11 at 15:12

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